mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (last s) x₁
hsat₂ : IsMaximal (last s) x₂
hsaty₁ : IsMaximal (snoc s x₁ hsat₁).last y₁
hsaty₂ : IsMaximal (snoc s x₂ hsat₂).last y₂
hr₁ : Iso (last s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (last s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) := Equiv.swap (Fin.last (s.length + 1)) (Fin.last s.length).castSucc
h1 : ∀ {i : Fin s.length}, i.castSucc.castSucc ≠ (Fin.last s.length).castSucc
h2 : ∀ {i : Fin s.length}, i.castSucc.castSucc ≠ Fin.last (s.length + 1)
⊢ ∀ (i : Fin ((snoc s x₁ hsat₁).snoc y₁ hsaty₁).length),
    Iso (((snoc s x₁ hsat₁).snoc y₁ hsaty₁).toFun i.castSucc, ((snoc s x₁ hsat₁).snoc y₁ hsaty₁).toFun i.succ)
      (((snoc s x₂ hsat₂).snoc y₂ hsaty₂).toFun (e i).castSucc, ((snoc s x₂ hsat₂).snoc y₂ hsaty₂).toFun (e i).succ)

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hs₁ : s₁.length = 0
⊢ ↑0 = ↑(Fin.last s₁.length)

Tactic:
simp [hs₁]

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hs₁ : s₁.length = 0
⊢ ↑0 = ↑(Fin.last s₁.length)

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hs₁ : s₁.length = 0
⊢ s₂.length = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: ι✝ : Type u_1
R✝ : Type u_2
S : Type u_3
inst✝⁸ : Ring R✝
inst✝⁷ : Ring S
M : Type u_4
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R✝ M
m : Submodule R✝ M
N : Type u_5
inst✝⁴ : AddCommGroup N
inst✝³ : Module R✝ N
ι : Type u_7
inst✝² : Finite ι
R : ι → Type u_6
inst✝¹ : (i : ι) → Ring (R i)
inst✝ : ∀ (i : ι), IsSemisimpleRing (R i)
⊢ IsSemisimpleRing ((i : ι) → R i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
⊢ ¬0 < s₂.length → ¬0 < s₁.length

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hs₁0 : s₁.length = 0
⊢ s₁ = s₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
⊢ (eraseLast s).head ≤ x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : Ring S
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
m : Submodule R M
N : Type u_5
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hR : IsSemisimpleRing R
hS : IsSemisimpleRing S
⊢ IsSemisimpleRing (R × S)

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
⊢ (eraseLast s).length = n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
hf : Function.Surjective ⇑f
inst✝ : IsSemisimpleRing R
⊢ IsSemisimpleRing S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : IsSemisimpleRing R
J : Ideal R
f : Module.End R R
idem : IsIdempotentElem f
h : IsCompl (LinearMap.range f) J
⊢ LinearMap.range f = Ideal.span {f 1}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : IsSemisimpleRing R
I : Ideal R
⊢ ∃ e, IsIdempotentElem e ∧ I = Ideal.span {e}

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
⊢ (eraseLast s).last ⊓ x = last t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
M : Type u_4
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_5
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsSimpleModule R M
f : M →ₗ[R] N
⊢ Function.Injective ⇑f ∨ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ torsionOf R M 0 = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
h : m = 0
⊢ torsionOf R M m = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
M : Type u_4
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_5
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsSimpleModule R N
f : M →ₗ[R] N
⊢ Function.Surjective ⇑f ∨ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ torsionOf R M m = ⊤ ↔ m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
M : Type u_4
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_5
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsSimpleModule R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
⊢ IsCoatom f.ker

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁷ : Ring R
inst✝⁶ : Ring S
M : Type u_4
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
m : Submodule R M
N : Type u_5
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : DecidableEq (Module.End R M)
inst✝ : IsSimpleModule R M
a : Module.End R M
a0 : a ≠ 0
⊢ (a * if h : a = 0 then 0 else ↑(LinearEquiv.ofBijective a ⋯).symm) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
m : M
⊢ torsionOf R M m = ⊥ ↔ m ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
⊢ (R ∙ x).annihilator = torsionOf R M x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: ι : Type u_1
R✝ : Type u_2
S : Type u_3
inst✝⁸ : Ring R✝
inst✝⁷ : Ring S
M : Type u_4
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R✝ M
m : Submodule R✝ M
N : Type u_5
inst✝⁴ : AddCommGroup N
inst✝³ : Module R✝ N
R : Type u_6
inst✝² : DivisionRing R
inst✝¹ : Module R M
inst✝ : Nontrivial M
v : M
hv : v ≠ 0
w : M
f : M →ₗ[R] M
eq : f ∘ₗ toSpanSingleton R M v = toSpanSingleton R M w
⊢ (toSpanSingleton (Module.End R M) M v) f = w

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ annihilator R M = ⨅ x, torsionOf R M x

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
hmtx : IsMaximal (last t) x
⊢ (snoc t x hmtx).head = head s

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
hmtx : IsMaximal (last t) x
⊢ (snoc t x hmtx).length + 1 = n + 1

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
hmtx : IsMaximal (last t) x
⊢ (snoc t x hmtx).last = x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : Ring S
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
m : Submodule R M
N : Type u_5
inst✝¹ : AddCommGroup N
inst✝ : Module R N
X Y : Submodule R M × Submodule R M
h✝ : Iso X Y
e : Submodule R M × Submodule R M → Submodule R M × Submodule R M → Prop :=
  fun X Y ↦ Nonempty ((↥X.2 ⧸ Submodule.comap X.2.subtype X.1) ≃ₗ[R] ↥Y.2 ⧸ Submodule.comap Y.2.subtype Y.1)
x✝ y✝ : Submodule R M
h : IsMaximal x✝ (x✝ ⊔ y✝)
⊢ e (x✝, x✝ ⊔ y✝) (x✝ ⊓ y✝, y✝)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: ι : Type u_3
R : Type u_4
M : Type u_5
v : ι → M
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥
i : ι
r : R
hi : r • v i ∈ Submodule.span R (v '' (Set.univ \ {i}))
hv : R ∙ v i ⊓ Submodule.span R (Set.range fun i_1 ↦ v ↑i_1) = ⊥
⊢ r • v i ∈ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: ι : Type u_3
R : Type u_4
M : Type u_5
v : ι → M
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
hv : iSupIndep fun i ↦ R ∙ v i
h_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥
⊢ LinearIndependent R v

Tactic:
by

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
hmtx : IsMaximal (last t) x
⊢ ((snoc t (eraseLast s).last ⋯).last, last s) ∈ {(x, y) | IsMaximal x y}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
S : Type u_3
inst✝² : CommMonoid S
inst✝¹ : DistribMulAction S M
inst✝ : SMulCommClass S R M
⊢ ∀ {a b : M}, a ∈ {x | ∃ a, a • x = 0} → b ∈ {x | ∃ a, a • x = 0} → a + b ∈ {x | ∃ a, a • x = 0}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
S : Type u_3
inst✝² : CommMonoid S
inst✝¹ : DistribMulAction S M
inst✝ : SMulCommClass S R M
a : R
x : M
x✝ : x ∈ {x | ∃ a, a • x = 0}
b : S
h : b • x = 0
⊢ b • a • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ IsTorsionBy R M a ↔ a ∈ annihilator R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
f✝ : End (End R M) M
s : Finset M
x : ↥s →₀ M := Finsupp.equivFunOnFinite.symm fun x ↦ ↑x
w✝ : Submodule R (↥s →₀ M)
h : IsCompl (R ∙ x) w✝
p : (↥s →₀ M) →ₗ[R] ↥s →₀ M := (R ∙ x).projection w✝ h
f : End (End R (↥s →₀ M)) (↥s →₀ M) := (End.ringHomEndFinsupp ↥s) f✝
this : f (p • x) = f x
⊢ f x ∈ R ∙ x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
f✝ : End (End R M) M
s : Finset M
x : ↥s →₀ M := Finsupp.equivFunOnFinite.symm fun x ↦ ↑x
w✝ : Submodule R (↥s →₀ M)
h : IsCompl (R ∙ x) w✝
p : (↥s →₀ M) →ₗ[R] ↥s →₀ M := (R ∙ x).projection w✝ h
f : End (End R (↥s →₀ M)) (↥s →₀ M) := (End.ringHomEndFinsupp ↥s) f✝
this✝ : f (p • x) = f x
this : f x ∈ R ∙ x
r : R
hr : r • x = f x
m : M
hm : m ∈ s
⊢ f✝ m = r • m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ IsTorsionBySet R M s ↔ s ⊆ ↑(annihilator R M)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
x : M
⊢ x ∈ torsionBySet R M s ↔ ∀ (a : ↑s), ↑a • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ torsionBySet R M {a} = torsionBy R M a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
x : M
hx : ∀ (a : ↑s), ↑a • x = 0
this : Ideal.span s ≤ Ideal.torsionOf R M x
⊢ ∀ (a : ↑↑(Ideal.span s)), ↑a • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ torsionBySet R M s = torsionBySet R M ↑(Ideal.span s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ torsionBySet R M s = torsionBySet R M ↑(Ideal.span s)

Tactic:
by
  refine le_antisymm (fun x hx => ?_) (torsionBySet_le_torsionBySet_of_subset subset_span)
  rw [mem_torsionBySet_iff] at hx ⊢
  suffices Ideal.span s ≤ Ideal.torsionOf R M x by
    rintro ⟨a, ha⟩
    exact this ha
  rw [Ideal.span_le]
  exact fun a ha => hx ⟨a, ha⟩

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a b : R
dvd : a ∣ b
⊢ torsionBy R M a ≤ torsionBy R M b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x✝ : M
h : x✝ ∈ torsionBy R M 1
⊢ x✝ ∈ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SimpleModule.Basic

Goal state before:
pp: R : Type u_2
inst✝⁴ : Ring R
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsSemisimpleModule R M
inst✝ : Module.Finite (End R M) M
⊢ Function.Surjective ⇑(toModuleEnd (End R M) M)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ torsionBySet R M Set.univ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ IsTorsionBySet R M {a} ↔ IsTorsionBy R M a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ IsTorsionBySet R M s ↔ IsTorsionBySet R M ↑(Ideal.span s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
h : torsionBySet R M s = ⊤
x : M
⊢ ∀ ⦃a : ↑s⦄, ↑a • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
t : R
m : AEval R M a
⊢ (of R M a).symm (C t • m) = (of R M a).symm (t • m)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_2
A : Type u_3
M : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : M
⊢ X • (of R M a) m = (of R M a) (a • m)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_2
A : Type u_3
M : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : M
⊢ X • (of R M a) m = (of R M a) (a • m)

Tactic:
by
  rw [← of_aeval_smul, aeval_X]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_2
A : Type u_3
M : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ X ^ n • (of R M a) m = (of R M a) (a ^ n • m)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : AEval R M a
⊢ (of R M a).symm (X • m) = a • (of R M a).symm m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ IsTorsionBy R M a ↔ torsionBy R M a = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
I : Ideal R
hbot : I ≠ ⊥
⊢ IsTorsion R ↥(torsionBySet R M ↑I)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ (r • f) • m = r • f • m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type ?u.5
A : Type ?u.7
M : Type ?u.13
inst✝¹⁰ : CommSemiring R
inst✝⁹ : Semiring A
a : A
inst✝⁸ : Algebra R A
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module A M
inst✝⁵ : Module R M
inst✝⁴ : IsScalarTower R A M
N : Type ?u.26
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module R[X] N
inst✝ : IsScalarTower R R[X] N
f : M →ₗ[R] N
hf : ∀ (m : M), f (a • m) = X • f m
p : R[X]
k : R
m : AEval R M a
⊢ (f ∘ₗ ↑(of R M a).symm).toFun (C k • m) = (RingHom.id R[X]) (C k) • (f ∘ₗ ↑(of R M a).symm).toFun m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type ?u.5
A : Type ?u.7
M : Type ?u.13
inst✝¹⁰ : CommSemiring R
inst✝⁹ : Semiring A
a : A
inst✝⁸ : Algebra R A
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module A M
inst✝⁵ : Module R M
inst✝⁴ : IsScalarTower R A M
N : Type ?u.26
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module R[X] N
inst✝ : IsScalarTower R R[X] N
f : M →ₗ[R] N
hf : ∀ (m : M), f (a • m) = X • f m
p✝ p q : R[X]
hp : ∀ (x : AEval R M a), (f ∘ₗ ↑(of R M a).symm).toFun (p • x) = (RingHom.id R[X]) p • (f ∘ₗ ↑(of R M a).symm).toFun x
hq : ∀ (x : AEval R M a), (f ∘ₗ ↑(of R M a).symm).toFun (q • x) = (RingHom.id R[X]) q • (f ∘ₗ ↑(of R M a).symm).toFun x
m : AEval R M a
⊢ (f ∘ₗ ↑(of R M a).symm).toFun ((p + q) • m) = (RingHom.id R[X]) (p + q) • (f ∘ₗ ↑(of R M a).symm).toFun m

Tactic:
by

Goal state after: