mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Lie.Free

Goal state before:
pp: case mk.mk
R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
f : X → L
x✝ : FreeLieAlgebra R X
a : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y✝ : FreeLieAlgebra R X
b : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Quot.liftOn ⁅Quot.mk (Rel R X) a, Quot.mk (Rel R X) b⁆ ⇑(liftAux R f) ⋯ = (liftAux R f) (a * b)

Tactic:
]

Goal state after:
case mk.mk
R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
f : X → L
x✝ : FreeLieAlgebra R X
a : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y✝ : FreeLieAlgebra R X
b : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Quot.liftOn ⁅Quot.mk (Rel R X) a, Quot.mk (Rel R X) b⁆ ⇑(liftAux R f) ⋯ = (liftAux R f) (a * b)
Module: Mathlib.Algebra.Lie.Free

Goal state before:
pp: R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
f : X → L
x : X
⊢ f x = f x

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Lie.Free

Goal state before:
pp: R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
F : FreeLieAlgebra R X →ₗ⁅R⁆ L
⊢ (lift R) ((lift R).symm F) = F

Tactic:
]

Goal state after:
R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
F : FreeLieAlgebra R X →ₗ⁅R⁆ L
⊢ (lift R) ((lift R).symm F) = F
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
R : Type u_3
inst✝ : CommSemiring R
m : M
a : R[M]
⊢ a ∈ grade R m ↔ ↑a.coeff.support ⊆ {m}

Tactic:
]

Goal state after:
M : Type u_1
R : Type u_3
inst✝ : CommSemiring R
m : M
a : R[M]
⊢ a ∈ grade R m ↔ ↑a.coeff.support ⊆ {m}
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
R : Type u_3
inst✝ : CommSemiring R
m : M
a : R[M]
⊢ (∃ b, a.coeff = Finsupp.single m b) ↔ a ∈ (lsingle m).range

Tactic:
]

Goal state after:
M : Type u_1
R : Type u_3
inst✝ : CommSemiring R
m : M
a : R[M]
⊢ (∃ b, a.coeff = Finsupp.single m b) ↔ a ∈ (lsingle m).range
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : M → ι
m : M
r : R
x : M
hx : x ∈ (single m r).coeff.support
⊢ f m = f m

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝² : AddMonoid M
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
m : M
h : m ∈ (single 0 1).coeff.support
⊢ f m = 0

Tactic:
]

Goal state after:
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝² : AddMonoid M
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
m : M
h : m ∈ (single 0 1).coeff.support
⊢ f m = 0
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝² : AddMonoid M
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i j : ι
a b : R[M]
ha : a ∈ gradeBy R (⇑f) i
hb : b ∈ gradeBy R (⇑f) j
ma : M
hma : ma ∈ a.coeff.support
mb : M
hmb : mb ∈ b.coeff.support
hc : ma + mb ∈ (a * b).coeff.support
⊢ f ma + f mb = i + j

Tactic:
ha ma hma,

Goal state after:
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝² : AddMonoid M
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i j : ι
a b : R[M]
ha : a ∈ gradeBy R (⇑f) i
hb : b ∈ gradeBy R (⇑f) j
ma : M
hma : ma ∈ a.coeff.support
mb : M
hmb : mb ∈ b.coeff.support
hc : ma + mb ∈ (a * b).coeff.support
⊢ i + f mb = i + j
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝² : AddMonoid M
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i j : ι
a b : R[M]
ha : a ∈ gradeBy R (⇑f) i
hb : b ∈ gradeBy R (⇑f) j
ma : M
hma : ma ∈ a.coeff.support
mb : M
hmb : mb ∈ b.coeff.support
hc : ma + mb ∈ (a * b).coeff.support
⊢ i + j = i + j

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i j : Multiplicative M
⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f (Multiplicative.toAdd (i * j))))
      ⟨single (Multiplicative.toAdd (i * j)) 1, ⋯⟩ =
    (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f (Multiplicative.toAdd i))) ⟨single (Multiplicative.toAdd i) 1, ⋯⟩ *
      (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f (Multiplicative.toAdd j))) ⟨single (Multiplicative.toAdd j) 1, ⋯⟩

Tactic:
simpa [toAdd_mul, of_mul_of, GradedMonoid.GMul.mul, single_mul_single, mul_one] using
        DirectSum.of_eq_of_gradedMonoid_eq <| Sigma.subtype_ext (f.map_add _ _) rfl

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
m : M
r : R
⊢ single (Multiplicative.toAdd (Multiplicative.ofAdd m)) r = ↑(GradedMonoid.mk (f m) ⟨single m r, ⋯⟩).snd

Tactic:
]

Goal state after:
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
m : M
r : R
⊢ single (Multiplicative.toAdd (Multiplicative.ofAdd m)) r = ↑(GradedMonoid.mk (f m) ⟨single m r, ⋯⟩).snd
Module: Mathlib.Algebra.Lie.Graded

Goal state before:
pp: case hadd
ι : Type u_1
σ : Type u_2
R : Type u_3
L : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
ℒ : ι → Submodule R L
inst✝ : GradedLieAlgebra ℒ
φ : ι →+ R
y : ⨁ (i : ι), ↥(ℒ i)
hM :
  ∀ (k : ι) (b : ⨁ (i : ι), ↥(ℒ i)),
    (decompose ℒ).symm b ∈ ℒ k →
      (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) b = φ k • b
j i : ι
a : ↥(ℒ i)
f : ⨁ (i : ι), ↥(ℒ i)
a✝¹ : f i = 0
a✝ : a ≠ 0
ih :
  ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) ⁅f, y⁆) j) =
    ↑(⁅f, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) y⁆ j) -
      ↑(⁅y, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) f⁆ j)
⊢ ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id))
          ⁅(of (fun i ↦ ↥(ℒ i)) i) a, y⁆)
        j) =
    ↑(⁅(of (fun i ↦ ↥(ℒ i)) i) a,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) y⁆
          j) -
      ↑(⁅y, φ i • (of (fun x ↦ ↥(ℒ x)) i) a⁆ j)

Tactic:
]

Goal state after:
case hadd
ι : Type u_1
σ : Type u_2
R : Type u_3
L : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
ℒ : ι → Submodule R L
inst✝ : GradedLieAlgebra ℒ
φ : ι →+ R
y : ⨁ (i : ι), ↥(ℒ i)
hM :
  ∀ (k : ι) (b : ⨁ (i : ι), ↥(ℒ i)),
    (decompose ℒ).symm b ∈ ℒ k →
      (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) b = φ k • b
j i : ι
a : ↥(ℒ i)
f : ⨁ (i : ι), ↥(ℒ i)
a✝¹ : f i = 0
a✝ : a ≠ 0
ih :
  ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) ⁅f, y⁆) j) =
    ↑(⁅f, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) y⁆ j) -
      ↑(⁅y, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) f⁆ j)
⊢ ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id))
          ⁅(of (fun i ↦ ↥(ℒ i)) i) a, y⁆)
        j) =
    ↑(⁅(of (fun i ↦ ↥(ℒ i)) i) a,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) y⁆
          j) -
      ↑(⁅y, φ i • (of (fun x ↦ ↥(ℒ x)) i) a⁆ j)
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby✝ : single m b + y ∈ gradeBy R (⇑f) i
hmby : ↑(Finsupp.single m b).support ∪ ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
this : Disjoint (Finsupp.single m b).support y.coeff.support
⊢ (decomposeAux f) ↑⟨single m b + y, hmby✝⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨single m b + y, hmby✝⟩

Tactic:
Set.union_subset_iff

Goal state after:
case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby✝ : single m b + y ∈ gradeBy R (⇑f) i
hmby : ↑(Finsupp.single m b).support ⊆ ⇑f ⁻¹' {i} ∧ ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
this : Disjoint (Finsupp.single m b).support y.coeff.support
⊢ (decomposeAux f) ↑⟨single m b + y, hmby✝⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨single m b + y, hmby✝⟩
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby✝ : single m b + y ∈ gradeBy R (⇑f) i
hmby : ↑(Finsupp.single m b).support ⊆ ⇑f ⁻¹' {i} ∧ ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
this : Disjoint (Finsupp.single m b).support y.coeff.support
⊢ (decomposeAux f) ↑⟨single m b + y, hmby✝⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨single m b + y, hmby✝⟩

Tactic:
]

Goal state after:
case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby✝ : single m b + y ∈ gradeBy R (⇑f) i
hmby : ↑(Finsupp.single m b).support ⊆ ⇑f ⁻¹' {i} ∧ ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
this : Disjoint (Finsupp.single m b).support y.coeff.support
⊢ (decomposeAux f) ↑⟨single m b + y, hmby✝⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨single m b + y, hmby✝⟩
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby : single m b + y ∈ gradeBy R (⇑f) i
this : Disjoint (Finsupp.single m b).support y.coeff.support
h1 : m ∈ ⇑f ⁻¹' {i}
h2 : ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
⊢ f m = i

Tactic:
]

Goal state after:
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
i : ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
ih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨y, hx⟩
hmby : single m b + y ∈ gradeBy R (⇑f) i
this : Disjoint (Finsupp.single m b).support y.coeff.support
h1 : m ∈ ⇑f ⁻¹' {i}
h2 : ↑y.coeff.support ⊆ ⇑f ⁻¹' {i}
⊢ f m = i
Module: Mathlib.Algebra.Lie.Extension

Goal state before:
pp: R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x y z : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm (x + y)).1, ((ofProd c).symm z).1⁆,
        (↑↑c ((ofProd c).symm (x + y)).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm (x + y)).1, ((ofProd c).symm z).2⁆ -
          ⁅((ofProd c).symm z).1, ((ofProd c).symm (x + y)).2⁆) =
    (ofProd c)
      ((⁅((ofProd c).symm x).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm x).2⁆) +
        (⁅((ofProd c).symm y).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm y).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm y).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm y).2⁆))

Tactic:
]

Goal state after:
R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x y z : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm (x + y)).1, ((ofProd c).symm z).1⁆,
        (↑↑c ((ofProd c).symm (x + y)).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm (x + y)).1, ((ofProd c).symm z).2⁆ -
          ⁅((ofProd c).symm z).1, ((ofProd c).symm (x + y)).2⁆) =
    (ofProd c)
      ((⁅((ofProd c).symm x).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm x).2⁆) +
        (⁅((ofProd c).symm y).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm y).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm y).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm y).2⁆))
Module: Mathlib.Algebra.MonoidAlgebra.Grading

Goal state before:
pp: case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
this : Disjoint (Finsupp.single m b).support y.coeff.support
ih :
  ∀ (hx : y ∈ gradeBy R (⇑f) (f m)),
    (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨y, hx⟩
hmby : single m b + y ∈ gradeBy R (⇑f) (f m)
h1 : ↑(Finsupp.single m b).support ⊆ ⇑f ⁻¹' {f m}
h2 : ↑y.coeff.support ⊆ ⇑f ⁻¹' {f m}
ih' : (decomposeAux f) y = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨y, h2⟩ := ih h2
⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) (⟨single m b, ⋯⟩ + ⟨y, h2⟩) =
    (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨single m b + y, hmby⟩

Tactic:
]

Goal state after:
case refine_2
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝³ : AddMonoid M
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : CommSemiring R
f : M →+ ι
x : R[M]
m : M
b : R
y : R[M]
hmy : m ∉ y.coeff.support
hb : b ≠ 0
this : Disjoint (Finsupp.single m b).support y.coeff.support
ih :
  ∀ (hx : y ∈ gradeBy R (⇑f) (f m)),
    (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨y, hx⟩
hmby : single m b + y ∈ gradeBy R (⇑f) (f m)
h1 : ↑(Finsupp.single m b).support ⊆ ⇑f ⁻¹' {f m}
h2 : ↑y.coeff.support ⊆ ⇑f ⁻¹' {f m}
ih' : (decomposeAux f) y = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨y, h2⟩ := ih h2
⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) (⟨single m b, ⋯⟩ + ⟨y, h2⟩) =
    (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f m)) ⟨single m b + y, hmby⟩
Module: Mathlib.Algebra.Lie.Extension

Goal state before:
pp: R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x y z : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm x).1, ((ofProd c).symm (y + z)).1⁆,
        (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm (y + z)).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm (y + z)).2⁆ -
          ⁅((ofProd c).symm (y + z)).1, ((ofProd c).symm x).2⁆) =
    (ofProd c)
      ((⁅((ofProd c).symm x).1, ((ofProd c).symm y).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm y).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm y).2⁆ -
            ⁅((ofProd c).symm y).1, ((ofProd c).symm x).2⁆) +
        (⁅((ofProd c).symm x).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm x).2⁆))

Tactic:
]

Goal state after:
R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x y z : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm x).1, ((ofProd c).symm (y + z)).1⁆,
        (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm (y + z)).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm (y + z)).2⁆ -
          ⁅((ofProd c).symm (y + z)).1, ((ofProd c).symm x).2⁆) =
    (ofProd c)
      ((⁅((ofProd c).symm x).1, ((ofProd c).symm y).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm y).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm y).2⁆ -
            ⁅((ofProd c).symm y).1, ((ofProd c).symm x).2⁆) +
        (⁅((ofProd c).symm x).1, ((ofProd c).symm z).1⁆,
          (↑↑c ((ofProd c).symm x).1) ((ofProd c).symm z).1 + ⁅((ofProd c).symm x).1, ((ofProd c).symm z).2⁆ -
            ⁅((ofProd c).symm z).1, ((ofProd c).symm x).2⁆))
Module: Mathlib.Algebra.Lie.Extension

Goal state before:
pp: R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm x).1, ((ofProd c).symm x).1⁆,
        0 + ⁅((ofProd c).symm x).1, ((ofProd c).symm x).2⁆ - ⁅((ofProd c).symm x).1, ((ofProd c).symm x).2⁆) =
    (ofProd c) 0

Tactic:
]

Goal state after:
R : Type u_1
N : Type u_2
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
c : ↥(twoCocycle R L M)
x : ofTwoCocycle c
⊢ (ofProd c)
      (⁅((ofProd c).symm x).1, ((ofProd c).symm x).1⁆,
        0 + ⁅((ofProd c).symm x).1, ((ofProd c).symm x).2⁆ - ⁅((ofProd c).symm x).1, ((ofProd c).symm x).2⁆) =
    (ofProd c) 0
Module: Mathlib.LinearAlgebra.TensorProduct.Decomposition

Goal state before:
pp: case refine_1
ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ ((coeLinearMap fun i ↦ Submodule.baseChange S (ℳ i)) ∘ₗ lmap fun x ↦ Submodule.toBaseChange S (ℳ x)) ∘ₗ
      ↑(directSumRight R S S fun i ↦ ↥(ℳ i)) =
    LinearMap.id ∘ₗ ↑(LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ)).symm

Tactic:
]

Goal state after:
case refine_1
ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
⊢ ((coeLinearMap fun i ↦ Submodule.baseChange S (ℳ i)) ∘ₗ lmap fun x ↦ Submodule.toBaseChange S (ℳ x)) ∘ₗ
      ↑(directSumRight R S S fun i ↦ ↥(ℳ i)) =
    LinearMap.id ∘ₗ ↑(LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ)).symm
Module: Mathlib.LinearAlgebra.TensorProduct.Decomposition

Goal state before:
pp: case refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
i✝ : ι
⊢ ((((lmap fun x ↦ Submodule.toBaseChange S (ℳ x)) ∘ₗ
            ↑(directSumRight R S S fun i ↦ ↥(ℳ i)) ∘ₗ
              ↑(LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ))) ∘ₗ
          coeLinearMap fun i ↦ Submodule.baseChange S (ℳ i)) ∘ₗ
        lof S ι (fun i ↦ ↥(Submodule.baseChange S (ℳ i))) i✝) ∘ₗ
      Submodule.toBaseChange S (ℳ i✝) =
    (LinearMap.id ∘ₗ lof S ι (fun i ↦ ↥(Submodule.baseChange S (ℳ i))) i✝) ∘ₗ Submodule.toBaseChange S (ℳ i✝)

Tactic:
]

Goal state after:
case refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
i✝ : ι
⊢ ((((lmap fun x ↦ Submodule.toBaseChange S (ℳ x)) ∘ₗ
            ↑(directSumRight R S S fun i ↦ ↥(ℳ i)) ∘ₗ
              ↑(LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ))) ∘ₗ
          coeLinearMap fun i ↦ Submodule.baseChange S (ℳ i)) ∘ₗ
        lof S ι (fun i ↦ ↥(Submodule.baseChange S (ℳ i))) i✝) ∘ₗ
      Submodule.toBaseChange S (ℳ i✝) =
    (LinearMap.id ∘ₗ lof S ι (fun i ↦ ↥(Submodule.baseChange S (ℳ i))) i✝) ∘ₗ Submodule.toBaseChange S (ℳ i✝)
Module: Mathlib.Algebra.Lie.LieTheorem

Goal state before:
pp: R : Type u_1
L : Type u_2
A : Type u_3
V : Type u_4
inst✝¹⁹ : CommRing R
inst✝¹⁸ : IsPrincipalIdealRing R
inst✝¹⁷ : IsDomain R
inst✝¹⁶ : CharZero R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : LieRing A
inst✝¹² : LieAlgebra R A
inst✝¹¹ : Bracket L A
inst✝¹⁰ : Bracket A L
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module R V
inst✝⁷ : Module.Free R V
inst✝⁶ : Module.Finite R V
inst✝⁵ : LieRingModule L V
inst✝⁴ : LieModule R L V
inst✝³ : LieRingModule A V
inst✝² : LieModule R A V
inst✝¹ : IsLieTower L A V
inst✝ : IsLieTower A L V
χ : A → R
z : L
v : V
hv : ∀ (x : A), ⁅x, v⁆ = χ x • v
⊢ ∀ (x : A), ⁅x, ⁅z, v⁆⁆ = χ x • ⁅z, v⁆

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
A : Type u_3
V : Type u_4
inst✝¹⁹ : CommRing R
inst✝¹⁸ : IsPrincipalIdealRing R
inst✝¹⁷ : IsDomain R
inst✝¹⁶ : CharZero R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : LieRing A
inst✝¹² : LieAlgebra R A
inst✝¹¹ : Bracket L A
inst✝¹⁰ : Bracket A L
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module R V
inst✝⁷ : Module.Free R V
inst✝⁶ : Module.Finite R V
inst✝⁵ : LieRingModule L V
inst✝⁴ : LieModule R L V
inst✝³ : LieRingModule A V
inst✝² : LieModule R A V
inst✝¹ : IsLieTower L A V
inst✝ : IsLieTower A L V
χ : A → R
z : L
v : V
hv : ∀ (x : A), ⁅x, v⁆ = χ x • v
⊢ ∀ (x : A), ⁅x, ⁅z, v⁆⁆ = χ x • ⁅z, v⁆
Module: Mathlib.LinearAlgebra.TensorProduct.Decomposition

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
i : ι
x y : S ⊗[R] ↥(ℳ i)
h : (Submodule.toBaseChange S (ℳ i)) x = (Submodule.toBaseChange S (ℳ i)) y
⊢ Function.Bijective
    ⇑↑((LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ)).trans
          (directSumRight R S S fun i ↦ ↥(ℳ i)))

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
S : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
ℳ : ι → Submodule R M
inst✝³ : DecidableEq ι
inst✝² : Decomposition ℳ
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
i : ι
x y : S ⊗[R] ↥(ℳ i)
h : (Submodule.toBaseChange S (ℳ i)) x = (Submodule.toBaseChange S (ℳ i)) y
⊢ Function.Bijective
    ⇑↑((LinearEquiv.baseChange R S M (⨁ (i : ι), ↥(ℳ i)) (decomposeLinearEquiv ℳ)).trans
          (directSumRight R S S fun i ↦ ↥(ℳ i)))