module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Group.Subsemigroup.Membership | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 32
} | {
"line": 58,
"column": 2
} | [
{
"pp": "case refine_2\nM : Type u_2\ninst✝ : Mul M\nι : Sort u_3\nS : ι → Subsemigroup M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\nhx : x ∈ closure (⋃ i, ↑(S i))\n⊢ ∀ (x y : M),\n x ∈ closure (⋃ i, ↑(S i)) → y ∈ closure (⋃ i, ↑(S i)) → (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i",
"ppTerm": ... | [
"case refine_2\nM : Type u_2\ninst✝ : Mul M\nι : Sort u_3\nS : ι → Subsemigroup M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx✝ : M\nhx : x✝ ∈ closure (⋃ i, ↑(S i))\nx y : M\nhx✝ : x ∈ closure (⋃ i, ↑(S i))\nhy✝ : y ∈ closure (⋃ i, ↑(S i))\ni : ι\nhi : x ∈ S i\nj : ι\nhj : y ∈ S j\n⊢ ∃ i, x * y ∈ S i"
] | rintro x y _ _ ⟨i, hi⟩ ⟨j, hj⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Ring.Center | {
"line": 32,
"column": 57
} | {
"line": 32,
"column": 61
} | {
"line": 32,
"column": 62
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝ : NonAssocSemiring M\nx✝¹ x✝ : M\nn : ℕ\nihn : ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝\n⊢ ↑n * (x✝¹ * x✝) + x✝¹ * x✝ = (↑n + 1) * x✝¹ * x✝",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case succ\nM : Type u_1\ninst✝ : NonAssocSemiring M\nx✝¹ x✝ : M\nn : ℕ\nihn : ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝\n⊢ ↑n * x✝¹ * x✝ + x✝¹ * x✝ = (↑n + 1) * x✝¹ * x✝"
] | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Center | {
"line": 36,
"column": 48
} | {
"line": 36,
"column": 52
} | {
"line": 36,
"column": 53
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝ : NonAssocSemiring M\nx✝¹ x✝ : M\nn : ℕ\nihn : x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n)\n⊢ x✝¹ * x✝ * ↑n + x✝¹ * x✝ * 1 = x✝¹ * (x✝ * (↑n + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case succ\nM : Type u_1\ninst✝ : NonAssocSemiring M\nx✝¹ x✝ : M\nn : ℕ\nihn : x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n)\n⊢ x✝¹ * (x✝ * ↑n) + x✝¹ * x✝ * 1 = x✝¹ * (x✝ * (↑n + 1))"
] | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Prod | {
"line": 150,
"column": 15
} | {
"line": 150,
"column": 70
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_3\nT : Type u_5\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : NonUnitalNonAssocSemiring S\ninst✝ : NonUnitalNonAssocSemiring T\nf : R →ₙ+* S × T\nx : R\n⊢ (((fst S T).comp f).prod ((snd S T).comp f)) x = f x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": ... | [] | by simp only [prod_apply, coe_fst, coe_snd, comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Prod | {
"line": 226,
"column": 15
} | {
"line": 226,
"column": 70
} | {
"line": 228,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_3\nT : Type u_5\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring T\nf : R →+* S × T\nx : R\n⊢ (((fst S T).comp f).prod ((snd S T).comp f)) x = f x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"RingHom",
"Pr... | [] | by simp only [prod_apply, coe_fst, coe_snd, comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Submonoid.Basic | {
"line": 29,
"column": 25
} | {
"line": 29,
"column": 51
} | {
"line": 29,
"column": 52
} | [
{
"pp": "case add\nM : Type u_1\nR : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na b : R\nhb : b ∈ S\nr s : R\nhx✝ : r ∈ closure ↑S\nhy✝ : s ∈ closure ↑S\nhr : r * b ∈ closure ↑S\nhs : s * b ∈ closure ↑S\n⊢ (r + s) * b ∈ closure ↑S",
"ppTerm": "?add"... | [
"case add\nM : Type u_1\nR : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na b : R\nhb : b ∈ S\nr s : R\nhx✝ : r ∈ closure ↑S\nhy✝ : s ∈ closure ↑S\nhr : r * b ∈ closure ↑S\nhs : s * b ∈ closure ↑S\n⊢ r * b + s * b ∈ closure ↑S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Submonoid.Basic | {
"line": 38,
"column": 25
} | {
"line": 38,
"column": 51
} | {
"line": 38,
"column": 52
} | [
{
"pp": "case add\nM : Type u_1\nR : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na b : R\nha : a ∈ closure ↑S\nr s : R\nhx✝ : r ∈ closure ↑S\nhy✝ : s ∈ closure ↑S\nhr : a * r ∈ closure ↑S\nhs : a * s ∈ closure ↑S\n⊢ a * (r + s) ∈ closure ↑S",
"ppTerm... | [
"case add\nM : Type u_1\nR : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na b : R\nha : a ∈ closure ↑S\nr s : R\nhx✝ : r ∈ closure ↑S\nhy✝ : s ∈ closure ↑S\nhr : a * r ∈ closure ↑S\nhs : a * s ∈ closure ↑S\n⊢ a * r + a * s ∈ closure ↑S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 13
} | {
"line": 286,
"column": 14
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝² : Mul M\ninst✝¹ : Mul N\nι : Sort u_5\ninst✝ : Nonempty ι\nf : M →ₙ* N\nhf : Function.Injective ⇑f\ns : ι → Subsemigroup M\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"MulHom",
"Eq.mpr",
... | [
"M : Type u_1\nN : Type u_2\ninst✝² : Mul M\ninst✝¹ : Mul N\nι : Sort u_5\ninst✝ : Nonempty ι\nf : M →ₙ* N\nhf : Function.Injective ⇑f\ns : ι → Subsemigroup M\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 34
} | {
"line": 569,
"column": 35
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\ng : N →ₙ* P\nf : M →ₙ* N\n⊢ map g f.srange = (g.comp f).srange",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subsemigroup.map",
"congrArg",
"id",
"Mu... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\ng : N →ₙ* P\nf : M →ₙ* N\n⊢ map g (map f ⊤) = map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 37
} | {
"line": 752,
"column": 38
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M →ₙ* N\nS : Subsemigroup N\nh : S ≤ f.srange\n⊢ map f (comap f S) = S",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M →ₙ* N\nS : Subsemigroup N\nh : S ≤ f.srange\n⊢ map f (comap f S) = S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubsemiring.Basic | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 13
} | {
"line": 552,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁴ : NonUnitalNonAssocSemiring R\ninst✝³ : NonUnitalNonAssocSemiring S\nF : Type u_1\ninst✝² : FunLike F R S\ninst✝¹ : NonUnitalRingHomClass F R S\nι : Sort u_2\ninst✝ : Nonempty ι\nf : F\nhf : Function.Injective ⇑f\ns : ι → NonUnitalSubsemiring R\n⊢ ↑(map f (iInf s)) = ↑(⨅ ... | [
"R : Type u\nS : Type v\ninst✝⁴ : NonUnitalNonAssocSemiring R\ninst✝³ : NonUnitalNonAssocSemiring S\nF : Type u_1\ninst✝² : FunLike F R S\ninst✝¹ : NonUnitalRingHomClass F R S\nι : Sort u_2\ninst✝ : Nonempty ι\nf : F\nhf : Function.Injective ⇑f\ns : ι → NonUnitalSubsemiring R\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubsemiring.Basic | {
"line": 625,
"column": 27
} | {
"line": 625,
"column": 42
} | {
"line": 625,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocSemiring R\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubsemiring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) ⋯ (⨆ i, (S i).toAddSubmonoid) ⋯\nthis : ⨆ i, ... | [
"R : Type u\ninst✝ : NonUnitalNonAssocSemiring R\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubsemiring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) ⋯ (⨆ i, (S i).toAddSubmonoid) ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 34
} | {
"line": 175,
"column": 35
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring T\ng : S →+* T\nf : R →+* S\n⊢ Subsemiring.map g f.rangeS = (g.comp f).rangeS",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"R : Type u\nS : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring T\ng : S →+* T\nf : R →+* S\n⊢ Subsemiring.map g (Subsemiring.map f ⊤) = Subsemiring.map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 15
} | {
"line": 244,
"column": 16
} | [
{
"pp": "R : Type u\nS : Type v\nM : Type w\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : SMul R S\ninst✝ : IsScalarTower R S M\nx : R\nh : IsUnit ((algebraMap R (End S M)) x)\nm m' : M\nH : m = x • m'\n⊢... | [
"R : Type u\nS : Type v\nM : Type w\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : SMul R S\ninst✝ : IsScalarTower R S M\nx : R\nh : IsUnit ((algebraMap R (End S M)) x)\nm m' : M\nH : m = x • m'\n⊢ x • x • ↑h.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 15
} | {
"line": 253,
"column": 16
} | [
{
"pp": "R : Type u\nS : Type v\nM : Type w\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : SMul R S\ninst✝ : IsScalarTower R S M\nx : R\nh : IsUnit ((algebraMap R (End S M)) x)\nm m' : M\nH : m = x • m'\n⊢... | [
"R : Type u\nS : Type v\nM : Type w\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : SMul R S\ninst✝ : IsScalarTower R S M\nx : R\nh : IsUnit ((algebraMap R (End S M)) x)\nm m' : M\nH : m = x • m'\n⊢ x • m' = x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 33
} | {
"line": 238,
"column": 34
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type u_1\ninst✝² : NonUnitalNonAssocRing R\ninst✝¹ : NonUnitalNonAssocRing S\ninst✝ : NonUnitalNonAssocRing T\ng : S →ₙ+* T\nf : R →ₙ+* S\n⊢ NonUnitalSubring.map g f.range = (g.comp f).range",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u\nS : Type v\nT : Type u_1\ninst✝² : NonUnitalNonAssocRing R\ninst✝¹ : NonUnitalNonAssocRing S\ninst✝ : NonUnitalNonAssocRing T\ng : S →ₙ+* T\nf : R →ₙ+* S\n⊢ NonUnitalSubring.map g (NonUnitalSubring.map f ⊤) = NonUnitalSubring.map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 58
} | {
"line": 402,
"column": 59
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : CharZero R\n⊢ FaithfulSMul ℕ R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : CharZero R\n⊢ Injective ⇑(algebraMap ℕ R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 58
} | {
"line": 405,
"column": 59
} | [
{
"pp": "R✝ : Type u_1\nA : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R✝ A\ninst✝² : FaithfulSMul R✝ A\nR : Type u_3\ninst✝¹ : Ring R\ninst✝ : CharZero R\n⊢ FaithfulSMul ℤ R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebr... | [
"R✝ : Type u_1\nA : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R✝ A\ninst✝² : FaithfulSMul R✝ A\nR : Type u_3\ninst✝¹ : Ring R\ninst✝ : CharZero R\n⊢ Injective ⇑(algebraMap ℤ R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 476,
"column": 56
} | {
"line": 476,
"column": 67
} | {
"line": 476,
"column": 68
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nM : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\ninst✝⁷ : FaithfulSMul R A\ninst✝⁶ : Nontrivial R\ninst✝⁵ : IsCancelMulZero A\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module A M\ninst✝² : Module R M\ninst✝¹ : IsTorsionFree A M\ninst✝ : IsScalarTo... | [
"R : Type u_1\nA : Type u_3\nM : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\ninst✝⁷ : FaithfulSMul R A\ninst✝⁶ : Nontrivial R\ninst✝⁵ : IsCancelMulZero A\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module A M\ninst✝² : Module R M\ninst✝¹ : IsTorsionFree A M\ninst✝ : IsScalarTower R A M\nr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 13
} | {
"line": 612,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : R →+* S\nhf : Function.Injective ⇑f\ns : ι → Subsemiring R\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"R : Type u\nS : Type v\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : R →+* S\nhf : Function.Injective ⇑f\ns : ι → Subsemiring R\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 686,
"column": 27
} | {
"line": 686,
"column": 42
} | {
"line": 686,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : NonAssocSemiring R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : Subsemiring R := Subsemiring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubmonoid) ⋯ (⨆ i, (S i).toAddSubmonoid) ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i",
... | [
"R : Type u\ninst✝ : NonAssocSemiring R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : Subsemiring R := Subsemiring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubmonoid) ⋯ (⨆ i, (S i).toAddSubmonoid) ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 1051,
"column": 2
} | {
"line": 1051,
"column": 37
} | {
"line": 1051,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring S\nf : R →+* S\nt : Subsemiring S\nh : t ≤ f.rangeS\n⊢ map f (comap f t) = t",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring S\nf : R →+* S\nt : Subsemiring S\nh : t ≤ f.rangeS\n⊢ map f (comap f t) = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 534,
"column": 38
} | {
"line": 534,
"column": 59
} | {
"line": 534,
"column": 60
} | [
{
"pp": "case mul.add_left\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ z✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh₁ : x✝ * z✝ ∈ AddSubgroup.clo... | [
"case mul.add_left\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ z✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh₁ : x✝ * z✝ ∈ AddSubgroup.closure ↑(Subse... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 535,
"column": 39
} | {
"line": 535,
"column": 60
} | {
"line": 535,
"column": 61
} | [
{
"pp": "case mul.add_right\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ y✝ z✝ x✝ : R\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh₁ : x✝ * y✝ ∈ AddSubgroup.cl... | [
"case mul.add_right\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ y✝ z✝ x✝ : R\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh₁ : x✝ * y✝ ∈ AddSubgroup.closure ↑(Subs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 536,
"column": 30
} | {
"line": 536,
"column": 51
} | {
"line": 536,
"column": 52
} | [
{
"pp": "case mul.neg_left\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ -x✝ * y✝ ∈ AddSubgroup.closu... | [
"case mul.neg_left\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 537,
"column": 31
} | {
"line": 537,
"column": 52
} | {
"line": 537,
"column": 53
} | [
{
"pp": "case mul.neg_right\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x✝ * -y✝ ∈ AddSubgroup.clos... | [
"case mul.neg_right\nR : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x✝ * y✝ ∈ AddSubgroup.closure ↑(Subsemi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Hom | {
"line": 381,
"column": 20
} | {
"line": 381,
"column": 50
} | {
"line": 381,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nr : R\n⊢ (↑↑(algebraMap S A)).toFun ((algebraMap R S) r) = (algebraMap R A) r",
"ppTerm": "... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nr : R\n⊢ (algebraMap S A) ((algebraMap R S) r) = (algebraMap R A) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 616,
"column": 2
} | {
"line": 616,
"column": 13
} | {
"line": 616,
"column": 14
} | [
{
"pp": "F : Type w\nR : Type u\nS : Type v\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : FunLike F R S\ninst✝¹ : NonUnitalRingHomClass F R S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : F\nhf : Function.Injective ⇑f\ns : ι → NonUnitalSubring R\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i)... | [
"F : Type w\nR : Type u\nS : Type v\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S\ninst✝² : FunLike F R S\ninst✝¹ : NonUnitalRingHomClass F R S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : F\nhf : Function.Injective ⇑f\ns : ι → NonUnitalSubring R\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 690,
"column": 27
} | {
"line": 690,
"column": 42
} | {
"line": 690,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, ... | [
"R : Type u\ninst✝ : NonUnitalNonAssocRing R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, S i → ∃ i, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 710,
"column": 2
} | {
"line": 711,
"column": 35
} | {
"line": 711,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝¹ : NonUnitalNonAssocRing R\nι : Sort u_1\ninst✝ : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ IsMulCommutative ↥(⨆ i, S i)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonUni... | [
"R : Type u\ninst✝¹ : NonUnitalNonAssocRing R\nι : Sort u_1\ninst✝ : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ ∀ (a : R) (x : ι), a ∈ ↑(S x) → ∀ (a_1 : R) (x : ι), a_1 ∈ ↑(S x) → a * a_1 = a_1 * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Defs | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 47
} | {
"line": 67,
"column": 48
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\np : M\nhp : ¬IsUnit p\n⊢ Irreducible p ∨ ∃ a b, ¬IsUnit a ∧ ¬IsUnit b ∧ p = a * b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"HMul.hMul",
"eq_false",
"_private.Mathlib.Algebra.Group.Irreducible.D... | [
"M : Type u_1\ninst✝ : Monoid M\np : M\nhp : ¬IsUnit p\n⊢ (∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b) ∨ ∃ a, ¬IsUnit a ∧ ∃ x, ¬IsUnit x ∧ p = a * x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Defs | {
"line": 71,
"column": 24
} | {
"line": 71,
"column": 35
} | {
"line": 71,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\na b : M\ninst✝ : Subsingleton Mˣ\nhab : Irreducible (a * b)\n⊢ a = 1 ∨ b = 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : Monoid M\na b : M\ninst✝ : Subsingleton Mˣ\nhab : Irreducible (a * b)\n⊢ a = 1 ∨ b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Prime.Lemmas | {
"line": 49,
"column": 4
} | {
"line": 53,
"column": 15
} | {
"line": 53,
"column": 16
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : CommMonoidWithZero M\ninst✝⁴ : CommMonoidWithZero N\nF : Type u_3\nG : Type u_4\ninst✝³ : FunLike F M N\ninst✝² : MonoidWithZeroHomClass F M N\ninst✝¹ : FunLike G N M\ninst✝ : MulHomClass G N M\nf : F\ng : G\np : M\nhinv : ∀ (a : M), g (f a) = a\nhp : Prime (f p)\na... | [
"case refine_1\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommMonoidWithZero M\ninst✝⁴ : CommMonoidWithZero N\nF : Type u_3\nG : Type u_4\ninst✝³ : FunLike F M N\ninst✝² : MonoidWithZeroHomClass F M N\ninst✝¹ : FunLike G N M\ninst✝ : MulHomClass G N M\nf : F\ng : G\np : M\nhinv : ∀ (a : M), g (f a) = a\nhp : Prime (f p)... | refine
(hp.2.2 (f a) (f b) <| by
convert! map_dvd f h
simp).imp
?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Prime.Lemmas | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 69
} | {
"line": 140,
"column": 70
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : M\nhp : Prime p\na b : M\nk l : ℕ\nx✝² : p ^ k ∣ a\nx✝¹ : p ^ l ∣ b\nx✝ : p ^ (k + l + 1) ∣ a * b\nx : M\nhx : a = p ^ k * x\ny : M\nhy : b = p ^ l * y\nz : M\nhz : a * b = p ^ (k + l + 1) * z\n⊢ p ^ (k + l) * (x * y) = p ^ (k ... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : M\nhp : Prime p\na b : M\nk l : ℕ\nx✝² : p ^ k ∣ a\nx✝¹ : p ^ l ∣ b\nx✝ : p ^ (k + l + 1) ∣ a * b\nx : M\nhx : a = p ^ k * x\ny : M\nhy : b = p ^ l * y\nz : M\nhz : a * b = p ^ (k + l + 1) * z\n⊢ x * (y * (p ^ k * p ^ l)) = p * (z * (p ^ k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 41,
"column": 21
} | {
"line": 41,
"column": 32
} | {
"line": 41,
"column": 33
} | [
{
"pp": "M₀ : Type u_1\nS : Type u_2\ninst✝² : MonoidWithZero M₀\ninst✝¹ : SetLike S M₀\ninst✝ : SubmonoidClass S M₀\ns : S\nx : ↥s\nhx : Irreducible x\nh : ↑x = 0\n⊢ IsUnit x",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M₀ : Type u_1\nS : Type u_2\ninst✝² : MonoidWithZero M₀\ninst✝¹ : SetLike S M₀\ninst✝ : SubmonoidClass S M₀\ns : S\nx : ↥s\nhx : Irreducible x\nh : ↑x = 0\n⊢ IsUnit x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 59
} | {
"line": 190,
"column": 60
} | [
{
"pp": "F : Type u_1\nM₀ : Type u_2\nM₀' : Type u_3\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero M₀'\nr x y : M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : IsLeftCancelMulZero M₀\nz x✝¹ x✝ : ↥M₀⁰\nh : (fun x ↦ z * x) x✝¹ = (fun x ↦ z * x) x✝\n⊢ ↑z * ↑x✝¹ = ↑z * ↑x✝",
"ppTerm": "?m.25",
"assigned": false,... | [
"F : Type u_1\nM₀ : Type u_2\nM₀' : Type u_3\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero M₀'\nr x y : M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : IsLeftCancelMulZero M₀\nz x✝¹ x✝ : ↥M₀⁰\nh : (fun x ↦ z * x) x✝¹ = (fun x ↦ z * x) x✝\n⊢ ↑z * ↑x✝¹ = ↑z * ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 195,
"column": 6
} | {
"line": 195,
"column": 59
} | {
"line": 195,
"column": 60
} | [
{
"pp": "F : Type u_1\nM₀ : Type u_2\nM₀' : Type u_3\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero M₀'\nr x y : M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : IsRightCancelMulZero M₀\nz x✝¹ x✝ : ↥M₀⁰\nh : (fun x ↦ x * z) x✝¹ = (fun x ↦ x * z) x✝\n⊢ ↑x✝¹ * ↑z = ↑x✝ * ↑z",
"ppTerm": "?m.25",
"assigned": false... | [
"F : Type u_1\nM₀ : Type u_2\nM₀' : Type u_3\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero M₀'\nr x y : M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : IsRightCancelMulZero M₀\nz x✝¹ x✝ : ↥M₀⁰\nh : (fun x ↦ x * z) x✝¹ = (fun x ↦ x * z) x✝\n⊢ ↑x✝¹ * ↑z = ↑x✝ * ↑z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 33
} | {
"line": 260,
"column": 34
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type w\ninst✝² : NonAssocRing R\ninst✝¹ : NonAssocRing S\ninst✝ : NonAssocRing T\ng : S →+* T\nf : R →+* S\n⊢ Subring.map g f.range = (g.comp f).range",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subring.map",... | [
"R : Type u\nS : Type v\nT : Type w\ninst✝² : NonAssocRing R\ninst✝¹ : NonAssocRing S\ninst✝ : NonAssocRing T\ng : S →+* T\nf : R →+* S\n⊢ Subring.map g (Subring.map f ⊤) = Subring.map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 17
} | {
"line": 104,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝ : MonoidWithZero M\na : M\nh✝ : a ~ᵤ 0\nu : Mˣ\nh : 0 * ↑u = a\n⊢ a = 0",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : MonoidWithZero M\na : M\nh✝ : a ~ᵤ 0\nu : Mˣ\nh : 0 * ↑u = a\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 111,
"column": 57
} | {
"line": 111,
"column": 80
} | {
"line": 111,
"column": 81
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\na b : M\nu : Mˣ\nh : a * b * ↑u = 1\n⊢ a * (b * ↑u) = 1",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : CommMonoid M\na b : M\nu : Mˣ\nh : a * b * ↑u = 1\n⊢ a * (b * ↑u) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 27
} | {
"line": 199,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : IsLeftCancelMulZero M\na c : M\nhba : a * c ∣ a\n⊢ a ~ᵤ a * c",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"semigroupDvd",
"SemigroupWithZero.toSemigroup",
... | [
"M : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : IsLeftCancelMulZero M\na c d : M\na_eq : a = a * c * d\n⊢ a ~ᵤ a * c"
] | rcases hba with ⟨d, a_eq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 566,
"column": 38
} | {
"line": 566,
"column": 59
} | {
"line": 566,
"column": 60
} | [
{
"pp": "case mul.add_left\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ z✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh₁ : x✝ * z✝ ∈ AddSubgroup.closure ↑(Submonoid.c... | [
"case mul.add_left\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ z✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh₁ : x✝ * z✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 567,
"column": 39
} | {
"line": 567,
"column": 60
} | {
"line": 567,
"column": 61
} | [
{
"pp": "case mul.add_right\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ y✝ z✝ x✝ : R\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh₁ : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.... | [
"case mul.add_right\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ y✝ z✝ x✝ : R\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhz✝ : z✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh₁ : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 568,
"column": 30
} | {
"line": 568,
"column": 51
} | {
"line": 568,
"column": 52
} | [
{
"pp": "case mul.neg_left\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ -x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.clo... | [
"case mul.neg_left\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 569,
"column": 31
} | {
"line": 569,
"column": 52
} | {
"line": 569,
"column": 53
} | [
{
"pp": "case mul.neg_right\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x✝ * -y✝ ∈ AddSubgroup.closure ↑(Submonoid.cl... | [
"case mul.neg_right\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝¹ y✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy✝ : y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nh : x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x✝ * y✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 249,
"column": 2
} | {
"line": 251,
"column": 55
} | {
"line": 253,
"column": 0
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\nx y : M\n⊢ Prime x ∧ IsUnit y ∨ IsUnit x ∧ Prime y → Prime (x * y)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"HMul.hMul",
"MulZeroC... | [] | · rintro (⟨hx, hy⟩ | ⟨hx, hy⟩)
· exact (associated_mul_unit_left x y hy).symm.prime hx
· exact (associated_unit_mul_right y x hx).prime hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 33
} | {
"line": 167,
"column": 34
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₂₁ : R₂ →+* R\ninst✝ : RingHomSurjective τ₂₁\np : Submodule R M\nf : M₂ →ₛₗ[τ₂₁] M\nhf : ∀ (c : M₂), f c... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₂₁ : R₂ →+* R\ninst✝ : RingHomSurjective τ₂₁\np : Submodule R M\nf : M₂ →ₛₗ[τ₂₁] M\nhf : ∀ (c : M₂), f c ∈ p\n⊢ map ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 33
} | {
"line": 182,
"column": 34
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\n⊢ range 0 = ⊥",
"ppTerm": "?m.48",
"assigned": tru... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\n⊢ map 0 ⊤ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 556,
"column": 4
} | {
"line": 557,
"column": 27
} | {
"line": 558,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝ : CommMonoid M\na b : M\n⊢ (∃ x, a * x ~ᵤ b) → ∃ c, b = a * c",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Units.val",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"mul_assoc",
"Exists",
"Units",
... | [] | rintro ⟨x, u, rfl⟩
exact ⟨_, mul_assoc ..⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 556,
"column": 4
} | {
"line": 557,
"column": 27
} | {
"line": 558,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝ : CommMonoid M\na b : M\n⊢ (∃ x, a * x ~ᵤ b) → ∃ c, b = a * c",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Units.val",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"mul_assoc",
"Exists",
"Units",
... | [] | rintro ⟨x, u, rfl⟩
exact ⟨_, mul_assoc ..⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 13
} | {
"line": 699,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : NonAssocRing R\ninst✝¹ : NonAssocRing S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : R →+* S\nhf : Function.Injective ⇑f\ns : ι → Subring R\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"R : Type u\nS : Type v\ninst✝² : NonAssocRing R\ninst✝¹ : NonAssocRing S\nι : Sort u_1\ninst✝ : Nonempty ι\nf : R →+* S\nhf : Function.Injective ⇑f\ns : ι → Subring R\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 768,
"column": 27
} | {
"line": 768,
"column": 42
} | {
"line": 768,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : NonAssocRing R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : Subring R := Subring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubmonoid) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i",
"ppTerm": "?m.... | [
"R : Type u\ninst✝ : NonAssocRing R\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subring R\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : R\nU : Subring R := Subring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubmonoid) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\nthis : ⨆ i, S i ≤ U\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 15
} | {
"line": 659,
"column": 16
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝ : CommMonoidWithZero M\nx y : M\nu : Mˣ\nh : ∀ ⦃a b : M⦄, x * y * ↑u = a * b → IsUnit a ∨ IsUnit b\n⊢ IsUnit x ∨ IsUnit y",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nM : Type u_1\ninst✝ : CommMonoidWithZero M\nx y : M\nu : Mˣ\nh : ∀ ⦃a b : M⦄, x * y * ↑u = a * b → IsUnit a ∨ IsUnit b\n⊢ IsUnit x ∨ IsUnit y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 719,
"column": 2
} | {
"line": 720,
"column": 9
} | {
"line": 720,
"column": 10
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : M\nhp : Prime (Associates.mk p)\nm : M\nhle : Associates.mk m ≤ Associates.mk p\n⊢ Associates.mk m = 1 ∨ Associates.mk m = Associates.mk p",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : M\nhp : Prime (Associates.mk p)\nm : M\nhle : Associates.mk m ≤ Associates.mk p\n⊢ IsUnit m ∨ p ~ᵤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 222,
"column": 70
} | {
"line": 222,
"column": 95
} | {
"line": 222,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nM : Type u_10\nP : Type u_11\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nf : M →ₗ[R] P\ng : P →ₗ[R] M\nh : LeftInverse ⇑g ⇑f\nx : P\nhx : x ∈ (f ∘ₗ g - id).ker\n⊢ f (g x) = x",
"ppTerm": "?m.120",
"assigned": fal... | [
"R : Type u_1\ninst✝⁴ : Semiring R\nM : Type u_10\nP : Type u_11\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nf : M →ₗ[R] P\ng : P →ₗ[R] M\nh : LeftInverse ⇑g ⇑f\nx : P\nhx : x ∈ (f ∘ₗ g - id).ker\n⊢ f (g x) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 23
} | {
"line": 251,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : Ring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\ninst✝ : RingHomSurjective τ₁₂\np : Submodule R M\ny : M₂\nh₁ : y ∈ f.r... | [
"case mpr\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : Ring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\ninst✝ : RingHomSurjective τ₁₂\np : Submodule R M\ny : M₂\nh₁ : y ∈ f.range\nh₂ : ⇑... | rw [SetLike.le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 33
} | {
"line": 271,
"column": 34
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : CommSemiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nc : R₂\n⊢ (c • f).range ≤ f.range",... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : CommSemiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nc : R₂\n⊢ Submodule.map (c • f) ⊤ ≤ Submodule.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 33
} | {
"line": 282,
"column": 34
} | [
{
"pp": "K : Type u_4\nV : Type u_8\nV₂ : Type u_9\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\na : K\nh : a ≠ 0\n⊢ (a • f).range = f.range",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"K : Type u_4\nV : Type u_8\nV₂ : Type u_9\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\na : K\nh : a ≠ 0\n⊢ Submodule.map (a • f) ⊤ = Submodule.map f ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 33
} | {
"line": 286,
"column": 34
} | [
{
"pp": "K : Type u_4\nV : Type u_8\nV₂ : Type u_9\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\na : K\n⊢ (a • f).range = ⨆ (_ : a ≠ 0), f.range",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq... | [
"K : Type u_4\nV : Type u_8\nV₂ : Type u_9\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\na : K\n⊢ Submodule.map (a • f) ⊤ = ⨆ (_ : a ≠ 0), Submodule.map f ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 308,
"column": 50
} | {
"line": 308,
"column": 61
} | {
"line": 308,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ p.subtype.range = p",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ p.subtype.range = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 13
} | {
"line": 311,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\np' : Submodule R ↥p\n⊢ map p.subtype p' ≤ p",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\np' : Submodule R ↥p\n⊢ map p.subtype p' ≤ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 339,
"column": 14
} | {
"line": 339,
"column": 25
} | {
"line": 339,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q r : Submodule R M\nh : comap p.subtype q ≤ comap p.subtype r\n⊢ p ⊓ q ≤ p ⊓ r",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
... | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q r : Submodule R M\nh : comap p.subtype q ≤ comap p.subtype r\n⊢ p ⊓ q ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 340,
"column": 14
} | {
"line": 340,
"column": 25
} | {
"line": 340,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q r : Submodule R M\nh : p ⊓ q ≤ p ⊓ r\n⊢ comap p.subtype q ≤ comap p.subtype r",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q r : Submodule R M\nh : p ⊓ q ≤ p ⊓ r\n⊢ comap p.subtype q ≤ comap p.subtype r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Equiv | {
"line": 242,
"column": 21
} | {
"line": 242,
"column": 32
} | {
"line": 242,
"column": 33
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nN : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₗ[R] N\np : Submodule R N\nhf : Injective ⇑f\nh : ... | [
"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nN : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₗ[R] N\np : Submodule R N\nhf : Injective ⇑f\nh : p ≤ f.range\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Equiv | {
"line": 730,
"column": 15
} | {
"line": 730,
"column": 26
} | {
"line": 730,
"column": 27
} | [
{
"pp": "R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\ny : R\nx : A₁\nh : (algebraMap R A₂) y = e x\n⊢ (algebraMap R A₁) y = x",
"ppTerm": "?m.37",
"assigned": false,
"used... | [
"R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\ny : R\nx : A₁\nh : (algebraMap R A₂) y = e x\n⊢ (algebraMap R A₁) y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Equiv | {
"line": 882,
"column": 31
} | {
"line": 882,
"column": 62
} | {
"line": 883,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ne : R ≃ₗ[R] A\nx : R\n⊢ e.symm (e 1 * (algebraMap R A) x) = e.symm (x • e 1)",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | rw [Algebra.smul_def, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Equiv | {
"line": 882,
"column": 31
} | {
"line": 882,
"column": 62
} | {
"line": 883,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ne : R ≃ₗ[R] A\nx : R\n⊢ e.symm (e 1 * (algebraMap R A) x) = e.symm (x • e 1)",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | rw [Algebra.smul_def, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Equiv | {
"line": 882,
"column": 31
} | {
"line": 882,
"column": 62
} | {
"line": 883,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ne : R ≃ₗ[R] A\nx : R\n⊢ e.symm (e 1 * (algebraMap R A) x) = e.symm (x • e 1)",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | rw [Algebra.smul_def, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 1141,
"column": 2
} | {
"line": 1141,
"column": 37
} | {
"line": 1141,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : NonAssocRing R\ninst✝ : NonAssocRing S\nf : R →+* S\nt : Subring S\nh : t ≤ f.range\n⊢ map f (comap f t) = t",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝¹ : NonAssocRing R\ninst✝ : NonAssocRing S\nf : R →+* S\nt : Subring S\nh : t ≤ f.range\n⊢ map f (comap f t) = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 422,
"column": 2
} | {
"line": 434,
"column": 9
} | {
"line": 436,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass R S M\nr : S\nx : M\n⊢ x ∈ {r} • N ↔ ∃ m ∈ N, x = r • m",
"ppTerm": "?m.29",
"assigned": true,... | [] | fconstructor
· intro hx
induction x, hx using Submodule.set_smul_inductionOn with
| smul₀ => aesop
| @smul₁ t n mem h =>
rcases h with ⟨n, hn, rfl⟩
exact ⟨t • n, by aesop, smul_comm _ _ _⟩
| add mem₁ mem₂ h₁ h₂ =>
rcases h₁ with ⟨m₁, h₁, rfl⟩
rcases h₂ with ⟨m₂, h₂, rfl⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 422,
"column": 2
} | {
"line": 434,
"column": 9
} | {
"line": 436,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass R S M\nr : S\nx : M\n⊢ x ∈ {r} • N ↔ ∃ m ∈ N, x = r • m",
"ppTerm": "?m.29",
"assigned": true,... | [] | fconstructor
· intro hx
induction x, hx using Submodule.set_smul_inductionOn with
| smul₀ => aesop
| @smul₁ t n mem h =>
rcases h with ⟨n, hn, rfl⟩
exact ⟨t • n, by aesop, smul_comm _ _ _⟩
| add mem₁ mem₂ h₁ h₂ =>
rcases h₁ with ⟨m₁, h₁, rfl⟩
rcases h₂ with ⟨m₂, h₂, rfl⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ModularLattice | {
"line": 310,
"column": 33
} | {
"line": 310,
"column": 67
} | {
"line": 310,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsModularLattice α\na b c : α\nh₀ : Codisjoint a b\nh₁ : Disjoint b c\nh₂ : a ≤ c\n⊢ c ≤ a",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsModularLattice α\na b c : α\nh₀ : Codisjoint a b\nh₁ : Disjoint b c\nh₂ : a ≤ c\n⊢ c ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.ModularLattice | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 75
} | {
"line": 393,
"column": 76
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\na b c : α\ninst✝¹ : BoundedOrder α\ninst✝ : ComplementedLattice α\nhb : a ≤ b\nhc : b ≤ c\nd : α\nhdisjoint : Disjoint b d\nhcodisjoint : Codisjoint b d\n⊢ b ⊓ ((d ⊔ a) ⊓ c) = a",
"ppTerm": "?refine_1",
"assigned": tr... | [
"case refine_1\nα : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\na b c : α\ninst✝¹ : BoundedOrder α\ninst✝ : ComplementedLattice α\nhb : a ≤ b\nhc : b ≤ c\nd : α\nhdisjoint : Disjoint b d\nhcodisjoint : Codisjoint b d\n⊢ a ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Closure | {
"line": 207,
"column": 64
} | {
"line": 207,
"column": 79
} | {
"line": 207,
"column": 80
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\nc₁ c₂ : ClosureOperator α\nh : ∀ (x : α), c₁.IsClosed x ↔ c₂.IsClosed x\nx : α\n⊢ IsGLB {y | x ≤ y ∧ c₁.IsClosed y} (c₂ x)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
... | [
"α : Type u_1\ninst✝ : PartialOrder α\nc₁ c₂ : ClosureOperator α\nh : ∀ (x : α), c₁.IsClosed x ↔ c₂.IsClosed x\nx : α\n⊢ IsGLB {y | x ≤ y ∧ c₂.IsClosed y} (c₂ x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 189,
"column": 29
} | {
"line": 189,
"column": 56
} | {
"line": 189,
"column": 57
} | [
{
"pp": "case smul.add\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nx : M\nr₁ : R\ny x✝ y✝ : M\nhx✝ : x✝ ∈ closure (univ • s)\nhy✝ : y✝ ∈ closure (univ • s)\nh₁ : r₁ • x✝ ∈ closure (univ • s)\nh₂ : r₁ • y✝ ∈ closure (univ • s)\n⊢ r₁ • (x✝ + y✝) ∈ clo... | [
"case smul.add\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nx : M\nr₁ : R\ny x✝ y✝ : M\nhx✝ : x✝ ∈ closure (univ • s)\nhy✝ : y✝ ∈ closure (univ • s)\nh₁ : r₁ • x✝ ∈ closure (univ • s)\nh₂ : r₁ • y✝ ∈ closure (univ • s)\n⊢ r₁ • x✝ + r₁ • y✝ ∈ closure (uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 206,
"column": 16
} | {
"line": 206,
"column": 38
} | {
"line": 206,
"column": 39
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\np : (x : M) → x ∈ span R s → Prop\nzero : p 0 ⋯\nadd : ∀ (x y : M) (hx : x ∈ span R s) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯\nsmul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯\nx :... | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\np : (x : M) → x ∈ span R s → Prop\nzero : p 0 ⋯\nadd : ∀ (x y : M) (hx : x ∈ span R s) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯\nsmul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯\nx : M\nhx : x ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 13
} | {
"line": 309,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Submodule R M)\n⊢ span R (⋃ S ∈ s, ↑S) = sSup s",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Submodule R M)\n⊢ span R (⋃ S ∈ s, ↑S) = sSup s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupClosed | {
"line": 77,
"column": 21
} | {
"line": 77,
"column": 42
} | {
"line": 77,
"column": 43
} | [
{
"pp": "F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ns : Set α\ninst✝¹ : FunLike F β α\ninst✝ : SupHomClass F β α\nhs : SupClosed s\nf : F\na : β\nha : a ∈ ⇑f ⁻¹' s\nb : β\nhb : b ∈ ⇑f ⁻¹' s\n⊢ a ⊔ b ∈ ⇑f ⁻¹' s",
"ppTerm": "?m.15",
"assigned": true,
... | [
"F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ns : Set α\ninst✝¹ : FunLike F β α\ninst✝ : SupHomClass F β α\nhs : SupClosed s\nf : F\na : β\nha : a ∈ ⇑f ⁻¹' s\nb : β\nhb : b ∈ ⇑f ⁻¹' s\n⊢ f a ⊔ f b ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupClosed | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\n⊢ SupClosed (range ⇑f)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\n⊢ SupClosed (range ⇑f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupClosed | {
"line": 293,
"column": 8
} | {
"line": 293,
"column": 38
} | {
"line": 293,
"column": 39
} | [
{
"pp": "case refine_1\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [
"case refine_1\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ ↑u ×ˢ ↑v ⊆ s ×ˢ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 39
} | {
"line": 455,
"column": 4
} | [
{
"pp": "case refine_3\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : x ∈ R ∙ y\na b : R\n⊢ ∃ a_1, a_1 • y = a • y + b • y",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"DistribMu... | [] | exact ⟨a + b, by simp [add_smul]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Atoms | {
"line": 438,
"column": 2
} | {
"line": 439,
"column": 71
} | {
"line": 439,
"column": 72
} | [
{
"pp": "α : Type u_4\ninst✝ : Preorder α\n⊢ IsStronglyAtomic αᵒᵈ ↔ IsStronglyCoatomic α",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"_private.Mathlib.Order.Atoms.0.isStronglyAtomic_dual_iff_is_stronglyCoatomic._simp_1_1",
"Preorder.... | [
"α : Type u_4\ninst✝ : Preorder α\n⊢ (∀ (a a_1 : α), a_1 < a → ∃ a_3, a_3 ⋖ a ∧ a_1 ≤ a_3) ↔ ∀ (a b : α), a < b → ∃ x, x ⋖ b ∧ a ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 532,
"column": 13
} | {
"line": 532,
"column": 24
} | {
"line": 532,
"column": 25
} | [
{
"pp": "α : Type u_4\ninst✝¹ : BooleanAlgebra α\ninst✝ : IsAtomic α\nx y : α\nh : ∀ (a : α), IsAtom a → a ≤ x → a ≤ y\nhbot : ¬x ⊓ yᶜ = ⊥\na : α\nha : IsAtom a\nhle : a ≤ x ⊓ yᶜ\nhx : a ≤ x\nhy' : a ≤ yᶜ\nhy : a ≤ y\nthis : a ≤ y ⊓ yᶜ\n⊢ a = ⊥",
"ppTerm": "?m.85",
"assigned": false,
"usedConstants"... | [
"α : Type u_4\ninst✝¹ : BooleanAlgebra α\ninst✝ : IsAtomic α\nx y : α\nh : ∀ (a : α), IsAtom a → a ≤ x → a ≤ y\nhbot : ¬x ⊓ yᶜ = ⊥\na : α\nha : IsAtom a\nhle : a ≤ x ⊓ yᶜ\nhx : a ≤ x\nhy' : a ≤ yᶜ\nhy : a ≤ y\nthis : a ≤ y ⊓ yᶜ\n⊢ a = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 13
} | {
"line": 614,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : IsAtomistic α\ninst✝ : OrderTop α\n⊢ IsLUB {a | IsAtom a} ⊤",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : IsAtomistic α\ninst✝ : OrderTop α\n⊢ IsLUB {a | IsAtom a} ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupClosed | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 25
} | {
"line": 353,
"column": 26
} | [
{
"pp": "case sup\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\na b : α\nha : a ∈ f ⁻¹' latticeClosure (f '' s)\nhb : b ∈ f ⁻¹' latticeClosure (f '' s)\n⊢ a ⊔ b ∈ f ⁻¹' lattice... | [
"case sup\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\na b : α\nha : a ∈ f ⁻¹' latticeClosure (f '' s)\nhb : b ∈ f ⁻¹' latticeClosure (f '' s)\n⊢ f a ⊔ f b ∈ latticeClosure (f '' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupClosed | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 25
} | {
"line": 355,
"column": 26
} | [
{
"pp": "case left.inf\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\na b : α\nha : a ∈ f ⁻¹' latticeClosure (f '' s)\nhb : b ∈ f ⁻¹' latticeClosure (f '' s)\n⊢ a ⊓ b ∈ f ⁻¹' la... | [
"case left.inf\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\na b : α\nha : a ∈ f ⁻¹' latticeClosure (f '' s)\nhb : b ∈ f ⁻¹' latticeClosure (f '' s)\n⊢ f a ⊓ f b ∈ latticeClosure (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 15
} | {
"line": 706,
"column": 16
} | [
{
"pp": "case inl\nα : Type u_4\ninst✝ : CompleteAtomicBooleanAlgebra α\nS : Set α\nhS : ∀ a ∈ S, IsAtom a\nb : α\nhbS : b ∈ S\nx✝ : ⊥ ∈ {a | a ≤ sSup S ∧ IsAtom a}\nhale : ⊥ ≤ sSup S\nhatom : IsAtom ⊥\nhba : ⊥ ≤ b\n⊢ ⊥ ∈ S",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"case inl\nα : Type u_4\ninst✝ : CompleteAtomicBooleanAlgebra α\nS : Set α\nhS : ∀ a ∈ S, IsAtom a\nb : α\nhbS : b ∈ S\nx✝ : ⊥ ∈ {a | a ≤ sSup S ∧ IsAtom a}\nhale : ⊥ ≤ sSup S\nhatom : IsAtom ⊥\nhba : ⊥ ≤ b\n⊢ ⊥ ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 726,
"column": 4
} | {
"line": 726,
"column": 15
} | {
"line": 726,
"column": 16
} | [
{
"pp": "ι : Sort u_1\nα✝ : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α✝\nα : Type ?u.8\ninst✝ : CompleteAtomicBooleanAlgebra α\na b : α\n⊢ { toFun := fun A ↦ {a | ↑a ≤ A}, invFun := fun S ↦ sSup (Subtype.val '' S), left_inv := ⋯, right_inv := ⋯ } a ⊆\n { toFun := fun A ↦ {a | ↑a ≤ A}, invFun := fun S ... | [
"ι : Sort u_1\nα✝ : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α✝\nα : Type ?u.8\ninst✝ : CompleteAtomicBooleanAlgebra α\na b : α\n⊢ (∀ (a_1 : α), IsAtom a_1 → a_1 ≤ a → a_1 ≤ b) ↔ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 757,
"column": 20
} | {
"line": 757,
"column": 31
} | {
"line": 757,
"column": 32
} | [
{
"pp": "case inl.inl\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊥ ≠ ⊥\n⊢ ⊥ ≠ ⊤",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"id",
"Ne",
"Bot.bot",
"OrderTop.toTop",
"BoundedOrder.toOrderTo... | [
"case inl.inl\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊥ ≠ ⊥\n⊢ ¬⊥ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 757,
"column": 20
} | {
"line": 757,
"column": 31
} | {
"line": 757,
"column": 32
} | [
{
"pp": "case inr.inl\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊥\n⊢ ⊥ ≠ ⊤",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"id",
"Ne",
"Bot.bot",
"OrderTop.toTop",
"BoundedOrder.toOrderTo... | [
"case inr.inl\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊥\n⊢ ¬⊥ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 757,
"column": 20
} | {
"line": 757,
"column": 31
} | {
"line": 757,
"column": 32
} | [
{
"pp": "case inr.inr\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊤\n⊢ ⊥ ≠ ⊤",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"id",
"Ne",
"Bot.bot",
"OrderTop.toTop",
"BoundedOrder.toOrderTo... | [
"case inr.inr\nα : Type u_2\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊤\n⊢ ¬⊥ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 689,
"column": 4
} | {
"line": 689,
"column": 31
} | {
"line": 689,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nM₂ : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₂\nf : M →ₗ[R] M₂\nhf : Function.Surjective ⇑f\ninst✝ : IsPrincipal R M\nm : M\nhm : ⊤ = R ∙ m\nr : R\nx✝ : f (r • m) ∈ ⊤\n⊢ f (r • m) ∈ R ∙ f m",... | [
"R : Type u_1\nM : Type u_4\nM₂ : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₂\nf : M →ₗ[R] M₂\nhf : Function.Surjective ⇑f\ninst✝ : IsPrincipal R M\nm : M\nhm : ⊤ = R ∙ m\nr : R\nx✝ : f (r • m) ∈ ⊤\n⊢ r • f m ∈ R ∙ f m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 1186,
"column": 31
} | {
"line": 1186,
"column": 78
} | {
"line": 1186,
"column": 79
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : Lattice α\ninst✝³ : BoundedOrder α\ninst✝² : IsModularLattice α\ninst✝¹ : ComplementedLattice α\ninst✝ : IsAtomic α\na b : α\nhab : a < b\nha'b : ⊥ ≤ b\nha' : IsCompl ⟨a, ⋯⟩ ⟨⊥, ha'b⟩\n⊢ a = b",
"ppTerm": "?m.83",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"α : Type u_2\ninst✝⁴ : Lattice α\ninst✝³ : BoundedOrder α\ninst✝² : IsModularLattice α\ninst✝¹ : ComplementedLattice α\ninst✝ : IsAtomic α\na b : α\nhab : a < b\nha'b : ⊥ ≤ b\nha' : IsCompl ⟨a, ⋯⟩ ⟨⊥, ha'b⟩\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms | {
"line": 1190,
"column": 43
} | {
"line": 1190,
"column": 54
} | {
"line": 1190,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : Lattice α\ninst✝³ : BoundedOrder α\ninst✝² : IsModularLattice α\ninst✝¹ : ComplementedLattice α\ninst✝ : IsAtomic α\na b : α\nhab : a < b\na' : α\nha'b : a' ≤ b\nha' : IsCompl ⟨a, ⋯⟩ ⟨a', ha'b⟩\nd : α\nhd : IsAtom d ∧ d ≤ a'\n⊢ a ⊓ a' = ⊥",
"ppTerm": "?m.217",
"assigned":... | [
"α : Type u_2\ninst✝⁴ : Lattice α\ninst✝³ : BoundedOrder α\ninst✝² : IsModularLattice α\ninst✝¹ : ComplementedLattice α\ninst✝ : IsAtomic α\na b : α\nhab : a < b\na' : α\nha'b : a' ≤ b\nha' : IsCompl ⟨a, ⋯⟩ ⟨a', ha'b⟩\nd : α\nhd : IsAtom d ∧ d ≤ a'\n⊢ a ⊓ a' = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.EqLocus | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 38
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf g : M →ₛₗ[τ₁₂] M₂\nr : R\nx : M\nhx : f x = g x\n⊢ f (r • x) = g (r • x)",
"ppTerm":... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf g : M →ₛₗ[τ₁₂] M₂\nr : R\nx : M\nhx : f x = g x\n⊢ τ₁₂ r • f x = τ₁₂ r • g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Field | {
"line": 28,
"column": 36
} | {
"line": 28,
"column": 60
} | {
"line": 28,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\nM : Type u_2\ninst✝² : DivisionSemiring 𝕜\ninst✝¹ : AddCommMonoid M\ninst✝ : Module 𝕜 M\nr : 𝕜\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ m₁ = m₂",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"𝕜 : Type u_1\nM : Type u_2\ninst✝² : DivisionSemiring 𝕜\ninst✝¹ : AddCommMonoid M\ninst✝ : Module 𝕜 M\nr : 𝕜\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ m₁ = m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupIndep | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 22
} | {
"line": 345,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\n⊢ iSupIndep t ↔ ∀ (i : ι), Disjoint (t i) (sSup (t '' {j | j ≠ i}))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iSupIndep",
"congrArg",
"iSup",
"PartialOrder.toPreorder",... | [
"α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\n⊢ iSupIndep t ↔ ∀ (i : ι), Disjoint (t i) (⨆ a ∈ {j | j ≠ i}, t a)"
] | simp_rw [sSup_image] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.SupIndep | {
"line": 422,
"column": 35
} | {
"line": 422,
"column": 55
} | {
"line": 422,
"column": 56
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\nht : iSupIndep t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\nthis : univ = {i | t i ≠ ⊥}\n⊢ Injective t",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\nht : iSupIndep t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\nthis : univ = {i | t i ≠ ⊥}\n⊢ Injective t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 306,
"column": 24
} | {
"line": 306,
"column": 44
} | {
"line": 306,
"column": 45
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nhs : sSupIndep s\ncontra : s.Infinite\nt : Finset α\nht₁ : ↑t ⊆ s\nht₂ : sSup s = t.sup id\nthis : (s \\ ↑(insert ⊥ t)).Infinite\nx : α\nhx₁ : x ∈ s\nhx₂ : x ∉ ↑(insert ⊥ t)\n⊢ x ≠ ⊥ ∧ x ∉ t",
"ppTerm": "?m.89",
"assi... | [
"α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nhs : sSupIndep s\ncontra : s.Infinite\nt : Finset α\nht₁ : ↑t ⊆ s\nht₂ : sSup s = t.sup id\nthis : (s \\ ↑(insert ⊥ t)).Infinite\nx : α\nhx₁ : x ∈ s\nhx₂ : x ∉ ↑(insert ⊥ t)\n⊢ ¬x = ⊥ ∧ x ∉ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 310,
"column": 6
} | {
"line": 310,
"column": 60
} | {
"line": 310,
"column": 61
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nhs : sSupIndep s\nt : Finset α\nht₁ : ↑t ⊆ s\nht₂ : sSup s = t.sup id\nx : α\nhx₀ : x ∈ s\nhx₁ : x ≠ ⊥\nhx₂ : x ∉ t\nthis : Disjoint x (sSup ((↑t ∪ {x}) \\ {x}))\n⊢ x ⊓ sSup s = ⊥",
"ppTerm": "?m.154",
"assigned": fal... | [
"α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nhs : sSupIndep s\nt : Finset α\nht₁ : ↑t ⊆ s\nht₂ : sSup s = t.sup id\nx : α\nhx₀ : x ∈ s\nhx₁ : x ≠ ⊥\nhx₂ : x ∉ t\nthis : Disjoint x (sSup ((↑t ∪ {x}) \\ {x}))\n⊢ x ⊓ sSup s = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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