mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case neg
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Tactic:
obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho

Goal state after:
case neg
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
n' : ℕ
bN' : Basis (Fin n') R ↥N'
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ (∑ x, if ij.1 = x ∧ ij.2 = k then b₂ x else 0 • b₂ x) = if ij.2 = k then b₂ ij.1 else 0

Tactic:
zero_smul,

Goal state after:
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ (∑ x, if ij.1 = x ∧ ij.2 = k then b₂ x else 0) = if ij.2 = k then b₂ ij.1 else 0
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
y : S
x : R
s : ↥M
eq : y * (algebraMap R S) ↑(x, s).2 = (algebraMap R S) (x, s).1
u : Rˣ
hu : ↑u = ↑s
⊢ ∃ a, (Algebra.ofId R S) a = y

Tactic:
use x * u.inv

Goal state after:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
y : S
x : R
s : ↥M
eq : y * (algebraMap R S) ↑(x, s).2 = (algebraMap R S) (x, s).1
u : Rˣ
hu : ↑u = ↑s
⊢ (Algebra.ofId R S) (x * u.inv) = y
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
obtain ⟨a, ha⟩ := IsLocalization.exists_of_eq (M := M₂) hc

Goal state after:
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
a : ↥M₂
ha : ↑a * (c * r) = ↑a * (c * s)
⊢ ∃ c, ↑c * x = ↑c * y
Module: Mathlib.Algebra.Algebra.Epi

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsEpi R A
a✝ b✝ a b : A
⊢ a • 1 ⊗ₜ[R] b = a • b ⊗ₜ[R] 1

Tactic:
(isEpi_iff_forall_one_tmul_eq R A).mp inferInstance b

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsEpi R A
a✝ b✝ a b : A
⊢ a • b ⊗ₜ[R] 1 = a • b ⊗ₜ[R] 1
Module: Mathlib.Algebra.Algebra.Epi

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsEpi R A
a✝ b✝ a b : A
⊢ a • 1 ⊗ₜ[R] b = a • b ⊗ₜ[R] 1

Tactic:
rw [(isEpi_iff_forall_one_tmul_eq R A).mp inferInstance b]

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommSemiring β
⊢ NonUnitalCommSemiring α

Tactic:
by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalCommSemiring _ <;> intros <;> exact e.apply_symm_apply _

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

Goal state before:
pp: R : Type uR
R' : Type u_1
S : Type uS
T : Type u_2
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : CommSemiring B
inst✝ : Algebra R B
a : A
b : B
x : A ⊗[R] B
x✝¹ : A
x✝ : B
⊢ a • b • x✝¹ ⊗ₜ[R] x✝ = b • a • x✝¹ ⊗ₜ[R] x✝

Tactic:
simp [Algebra.smul_def, right_algebraMap_apply, smul_tmul']

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a✝ : Nontrivial R
h : ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0
⊢ Module.rank R M = 0

Tactic:
← nonpos_iff_eq_zero,

Goal state after:
case mpr
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a✝ : Nontrivial R
h : ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0
⊢ Module.rank R M ≤ 0
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
⊢ Module.rank R M = 0 ↔ ∀ (x : M), x = 0

Tactic:
rank_eq_zero_iff,

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
⊢ (∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0) ↔ ∀ (x : M), x = 0
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
inst✝ : FiniteDimensional K ↥p
f : V →ₗ[K] V
h : ∀ x ∈ p, f x ∈ p
h' : Disjoint p f.ker
x : V
hx : x ∈ comap f p
y : V
hy : y ∈ p
hxy : (f.restrict h) ⟨y, hy⟩ = ⟨f x, hx⟩
⊢ x ∈ p ⊔ f.ker

Tactic:
replace hxy : f y = f x := by simpa [Subtype.ext_iff] using hxy

Goal state after:
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
inst✝ : FiniteDimensional K ↥p
f : V →ₗ[K] V
h : ∀ x ∈ p, f x ∈ p
h' : Disjoint p f.ker
x : V
hx : x ∈ comap f p
y : V
hy : y ∈ p
hxy : f y = f x
⊢ x ∈ p ⊔ f.ker
Module: Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition

Goal state before:
pp: K : Type u
V : Type v
inst✝⁴ : Ring K
inst✝³ : StrongRankCondition K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
this : Free K ↥⊤
⊢ Module.rank K V ≤ 1 ↔ Module.rank K ↥⊤ ≤ 1

Tactic:
rank_top

Goal state after:
K : Type u
V : Type v
inst✝⁴ : Ring K
inst✝³ : StrongRankCondition K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
this : Free K ↥⊤
⊢ Module.rank K V ≤ 1 ↔ Module.rank K V ≤ 1
Module: Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition

Goal state before:
pp: K : Type u
V : Type v
inst✝⁵ : Ring K
inst✝⁴ : StrongRankCondition K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : Free K V
inst✝ : Module.Finite K V
this : Nontrivial K
s : Type v
hs : Basis s K V
⊢ lift.{u, v} #V = lift.{v, u} #K ^ lift.{u, v} (Module.rank K V)

Tactic:
have : Finite s := by
    obtain ⟨t, ht⟩ := ‹Module.Finite K V›
    exact basis_finite_of_finite_spans t.finite_toSet ht hs

Goal state after:
K : Type u
V : Type v
inst✝⁵ : Ring K
inst✝⁴ : StrongRankCondition K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : Free K V
inst✝ : Module.Finite K V
this✝ : Nontrivial K
s : Type v
hs : Basis s K V
this : Finite s
⊢ lift.{u, v} #V = lift.{v, u} #K ^ lift.{u, v} (Module.rank K V)
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Submodule K V
w : V
hS : finrank K ↥S = 1
hw : w ∈ S
hw0 : w ≠ 0
⊢ S = K ∙ w

Tactic:
have : FiniteDimensional K S := Module.finite_of_finrank_pos (by lia)

Goal state after:
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Submodule K V
w : V
hS : finrank K ↥S = 1
hw : w ∈ S
hw0 : w ≠ 0
this : FiniteDimensional K ↥S
⊢ S = K ∙ w
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: case zero
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
x : S
⊢ eval₂ f x ↑0 = ↑0

Tactic:
simp only [eval₂_zero, Nat.cast_zero]

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: case mul_zero
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ : Pre R X
⊢ liftFun R X f (a✝ * 0) = liftFun R X f 0

pp: case add_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ + c✝) = liftFun R X f (b✝ + c✝)

pp: case add_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ + a✝¹) = liftFun R X f (c✝ + b✝)

pp: case mul_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ * c✝) = liftFun R X f (b✝ * c✝)

pp: case mul_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ * a✝¹) = liftFun R X f (c✝ * b✝)

Tactic:
· change _ * algebraMap _ _ _ = algebraMap _ _ _
        simp

Goal state after:
case add_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ + c✝) = liftFun R X f (b✝ + c✝)

case add_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ + a✝¹) = liftFun R X f (c✝ + b✝)

case mul_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ * c✝) = liftFun R X f (b✝ * c✝)

case mul_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ * a✝¹) = liftFun R X f (c✝ * b✝)
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
c : Set (E →ₛₗ.[σ] F)
hc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c
cne : c.Nonempty
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (domain '' c)
x : ↥(sSup (domain '' c))
⊢ ∃ x_1, ↑x ∈ (↑x_1).domain

Tactic:
have := (mem_sSup_of_directed (cne.image _) hdir).1 x.2

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
c : Set (E →ₛₗ.[σ] F)
hc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c
cne : c.Nonempty
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (domain '' c)
x : ↥(sSup (domain '' c))
this : ∃ y ∈ domain '' c, ↑x ∈ y
⊢ ∃ x_1, ↑x ∈ (↑x_1).domain
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
x✝ : X
⊢ ((lift R) f) (ι R x✝) = f x✝

Tactic:
ι_def,

Goal state after:
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
x✝ : X
⊢ ((lift R) f) (Quot.mk (Rel R X) (Pre.of x✝)) = f x✝
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
x : X
⊢ ((lift R) f) (ι R x) = f x

Tactic:
ι_def,

Goal state after:
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
x : X
⊢ ((lift R) f) (Quot.mk (Rel R X) (Pre.of x)) = f x
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p r : R[X]
n : ℕ
⊢ (↑n * p).comp r = ↑n * p.comp r

Tactic:
rw [← C_eq_natCast, C_mul_comp]

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case h
R : Type u_1
inst✝⁷ : Ring R
E : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module R E
F : Type u_5
inst✝⁴ : AddCommGroup F
M : Type u_7
inst✝³ : Monoid M
inst✝² : DistribMulAction M F
inst✝¹ : Module R F
inst✝ : SMulCommClass R M F
f : E →ₗ.[R] F
z : M
x_fst : E
x_snd : F
fst✝ : E
snd✝ : F
h : fst✝ = x_fst ∧ z • snd✝ = x_snd
y : ↥f.domain
hy : ↑y = (fst✝, snd✝).1
hx' : ↑f y = (fst✝, snd✝).2
⊢ ↑y = (fst✝, x_snd).1 ∧ ↑(z • f) y = (fst✝, x_snd).2

Tactic:
← h.2

Goal state after:
case h
R : Type u_1
inst✝⁷ : Ring R
E : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module R E
F : Type u_5
inst✝⁴ : AddCommGroup F
M : Type u_7
inst✝³ : Monoid M
inst✝² : DistribMulAction M F
inst✝¹ : Module R F
inst✝ : SMulCommClass R M F
f : E →ₗ.[R] F
z : M
x_fst : E
x_snd : F
fst✝ : E
snd✝ : F
h : fst✝ = x_fst ∧ z • snd✝ = x_snd
y : ↥f.domain
hy : ↑y = (fst✝, snd✝).1
hx' : ↑f y = (fst✝, snd✝).2
⊢ ↑y = (fst✝, z • snd✝).1 ∧ ↑(z • f) y = (fst✝, z • snd✝).2
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case h
R : Type u_1
inst✝⁴ : Ring R
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
x_fst : E
x_snd : F
fst✝ : E
snd✝ : F
h : fst✝ = x_fst ∧ -snd✝ = x_snd
y : ↥f.domain
hy : ↑y = (fst✝, snd✝).1
hx' : ↑f y = (fst✝, snd✝).2
⊢ ↑y = (fst✝, x_snd).1 ∧ ↑(-f) y = (fst✝, x_snd).2

Tactic:
← h.2

Goal state after:
case h
R : Type u_1
inst✝⁴ : Ring R
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
x_fst : E
x_snd : F
fst✝ : E
snd✝ : F
h : fst✝ = x_fst ∧ -snd✝ = x_snd
y : ↥f.domain
hy : ↑y = (fst✝, snd✝).1
hx' : ↑f y = (fst✝, snd✝).2
⊢ ↑y = (fst✝, -snd✝).1 ∧ ↑(-f) y = (fst✝, -snd✝).2
Module: Mathlib.Order.Filter.Extr

Goal state before:
pp: α : Type u
β : Type v
inst✝ : Preorder β
f : α → β
s : Set α
a : α
ha : a ∈ s
hfsa : IsMinOn f s a
⊢ IsGLB {x | ∃ x_1 ∈ s, f x_1 = x} (f a)

Tactic:
rw [isGLB_iff_le_iff]

Goal state after:
α : Type u
β : Type v
inst✝ : Preorder β
f : α → β
s : Set α
a : α
ha : a ∈ s
hfsa : IsMinOn f s a
⊢ ∀ (b : β), b ≤ f a ↔ b ∈ lowerBounds {x | ∃ x_1 ∈ s, f x_1 = x}
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: R : Type u_1
A : Type u_3
T : Type u_4
ι : Type u_6
inst✝⁶ : SemilatticeInf T
inst✝⁵ : OrderTop T
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : AddCommMonoid T
inst✝¹ : AddLeftMono T
inst✝ : AddRightMono T
degt : A → T
degt0 : 0 ≤ degt 0
degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)
s : Finset ι
f : ι → R[A]
⊢ ∑ i ∈ s, (f i).coeff.support.inf degt =
    (Multiset.map (fun f ↦ f.coeff.support.inf degt) (Multiset.map (fun i ↦ f i) s.val)).sum

Tactic:
by rw [Multiset.map_map]; rfl

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
q : S[X]
hp : p ≠ 0
hpq : p.degree < q.degree
⊢ p.natDegree < q.natDegree

Tactic:
by_cases hq : q = 0

Goal state after:
case pos
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
q : S[X]
hp : p ≠ 0
hpq : p.degree < q.degree
hq : q = 0
⊢ p.natDegree < q.natDegree

case neg
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
q : S[X]
hp : p ≠ 0
hpq : p.degree < q.degree
hq : ¬q = 0
⊢ p.natDegree < q.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: case neg
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
q : S[X]
hp : p ≠ 0
hpq : p.degree < q.degree
hq : ¬q = 0
⊢ p.natDegree < q.natDegree

Tactic:
degree_eq_natDegree hp,

Goal state after:
case neg
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
q : S[X]
hp : p ≠ 0
hpq : ↑p.natDegree < q.degree
hq : ¬q = 0
⊢ p.natDegree < q.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: case neg
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.natDegree < n
hp : ¬p = 0
⊢ p.degree < ↑n

Tactic:
degree_eq_natDegree hp,

Goal state after:
case neg
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.natDegree < n
hp : ¬p = 0
⊢ ↑p.natDegree < ↑n
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
⊢ (erase p.natDegree p).degree ≠ p.degree

Tactic:
degree_eq_natDegree hp,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
⊢ (erase p.natDegree p).degree ≠ ↑p.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: case neg
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.natDegree < q.natDegree
hp : ¬p = 0
⊢ p.degree < q.degree

Tactic:
degree_eq_natDegree hp,

Goal state after:
case neg
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.natDegree < q.natDegree
hp : ¬p = 0
⊢ ↑p.natDegree < q.degree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p ≠ 0
h : p.degree < q.degree
hq : q ≠ 0
⊢ p.natDegree < q.natDegree

Tactic:
degree_eq_natDegree hp,

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p ≠ 0
h : ↑p.natDegree < q.degree
hq : q ≠ 0
⊢ p.natDegree < q.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: case neg
R : Type u
inst✝ : Semiring R
p : R[X]
a : R
hp : p ≠ 0
hpd : 0 < p.degree
⊢ (p + C a).natDegree = p.natDegree

Tactic:
degree_eq_natDegree hp,

Goal state after:
case neg
R : Type u
inst✝ : Semiring R
p : R[X]
a : R
hp : p ≠ 0
hpd : 0 < ↑p.natDegree
⊢ (p + C a).natDegree = p.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff + q.leadingCoeff ≠ 0
HEq : p.degree = q.degree
hlt : q.degree > (p + q).degree
⊢ p.leadingCoeff + q.leadingCoeff = 0

Tactic:
rw [leadingCoeff, leadingCoeff, natDegree_eq_of_degree_eq HEq, ← coeff_add]

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff + q.leadingCoeff ≠ 0
HEq : p.degree = q.degree
hlt : q.degree > (p + q).degree
⊢ (p + q).coeff q.natDegree = 0
Module: Mathlib.Algebra.Polynomial.Degree.Support

Goal state before:
pp: R : Type u
inst✝ : Semiring R
c : R
n : ℕ
⊢ #(C c * X ^ n).support ≤ 1

Tactic:
rw [← card_singleton n]

Goal state after:
R : Type u
inst✝ : Semiring R
c : R
n : ℕ
⊢ #(C c * X ^ n).support ≤ #{n}
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
⊢ p.natDegree = 0 ↔ p.degree ≤ 0

Tactic:
← nonpos_iff_eq_zero,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
⊢ p.natDegree ≤ 0 ↔ p.degree ≤ 0
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ p.degree + q.degree ≤ (p * q).degree

Tactic:
degree_eq_natDegree hp,

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ ↑p.natDegree + q.degree ≤ (p * q).degree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ p.degree + q.degree = ↑p.natDegree + ↑q.natDegree

Tactic:
degree_eq_natDegree hp,

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ ↑p.natDegree + q.degree = ↑p.natDegree + ↑q.natDegree
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
hq : q.Monic
this : DecidableEq R := Classical.decEq R
H : p.leadingCoeff ≠ 0
⊢ (p * q).leadingCoeff = p.leadingCoeff

Tactic:
rw [leadingCoeff_mul', hq.leadingCoeff, mul_one]

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
hq : q.Monic
this : DecidableEq R := Classical.decEq R
H : p.leadingCoeff ≠ 0
⊢ p.leadingCoeff * q.leadingCoeff ≠ 0