mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ∃ x_1, ¬((algebraMap R R⟦Γ⟧) x).coeff x_1 = ((single a) 1).coeff x_1

Tactic:
refine ⟨a, ?_⟩

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ¬((algebraMap R R⟦Γ⟧) x).coeff a = ((single a) 1).coeff a
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case pos
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
k : K
x : M
hk : ¬k = 0
c : FiniteArchimedeanClass M
y : ↥(↑f).domain
hy : ↑y - x ∈ ball K c
heq : ↑(k • y) - k • x = k • (↑y - x)
hy' : ↑(k • y) - k • x ∈ ball K c
⊢ f.evalCoeff (k • x) c = k • f.evalCoeff x c

Tactic:
simp [f.evalCoeff_eq hy, f.evalCoeff_eq hy', LinearPMap.map_smul]

Goal state after:
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
y : ↥(↑f).domain
this : x - ↑y ≠ 0
xy : FiniteArchimedeanClass M := FiniteArchimedeanClass.mk (x - ↑y) this
v : M
hv : v ∈ seed.stratum xy
hyv : ↑y + v ∈ (↑f).domain
h✝ : f.eval x = ↑↑f y
h : True
hu : ∀ (h : 0 ≠ 0), xy < FiniteArchimedeanClass.mk 0 h
huv : 0 + v = x - ↑y
huv' : x - ↑y - v = 0
⊢ ArchimedeanClass.mk (x - ↑y) ≠ ⊤

Tactic:
by simpa

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Hom

Goal state before:
pp: M : Type u_3
N : Type u_4
inst✝⁷ : Preorder M
inst✝⁶ : Add M
inst✝⁵ : Sub M
inst✝⁴ : OrderedSub M
inst✝³ : PartialOrder N
inst✝² : Add N
inst✝¹ : Sub N
inst✝ : OrderedSub N
e : M ≃o N
h_add : ∀ (a b : M), e (a + b) = e a + e b
a b : M
e_add : M ≃+ N := { toEquiv := e.toEquiv, map_add' := h_add }
this : e (e.symm (e a) - e.symm (e b)) ≤ e (e.symm (e a - e b))
⊢ e (a - b) ≤ e a - e b

Tactic:
by simpa

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s t : Set α
hdisj : Disjoint s t
hstab : stabilizer G s ≤ stabilizer G t
hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t
⊢ stabilizer G (s ∪ t) ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t

Tactic:
by simpa

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s t : Set α
hdisj : Disjoint s t
hstab : stabilizer G s ≤ stabilizer G t
hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t
⊢ stabilizer G s ≤ stabilizer G s ⊓ stabilizer G t

Tactic:
by simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : K
hx : 0 < mk x
⊢ stdPart (x + y) = stdPart y

Tactic:
obtain hy | hy := le_or_gt 0 (mk y)

Goal state after:
case inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : K
hx : 0 < mk x
hy : 0 ≤ mk y
⊢ stdPart (x + y) = stdPart y

case inr
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : K
hx : 0 < mk x
hy : mk y < 0
⊢ stdPart (x + y) = stdPart y
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : K
hy : 0 < mk y
⊢ 0 < mk (-y)

Tactic:
by simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: case inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
f : ℝ →+*o K
x : K
hx : 0 ≤ mk x
a : ℤ
ha : ↑a < x
b : ℤ
hb : x < ↑b
⊢ stdPart x = sInf {r | x < f r}

Tactic:
have hn : {r | x < f r}.Nonempty := ⟨b, by simpa using hb⟩

Goal state after:
case inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
f : ℝ →+*o K
x : K
hx : 0 ≤ mk x
a : ℤ
ha : ↑a < x
b : ℤ
hb : x < ↑b
hn : {r | x < f r}.Nonempty
⊢ stdPart x = sInf {r | x < f r}
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
c : FiniteArchimedeanClass M
y' : M
hy'✝ : y' ∈ (f.extendFun hx).domain
hy' : y' ∈ (↑f).domain ⊔ K ∙ x
a : M
ha : a ∈ (↑f).domain
b : M
hb : b ∈ K ∙ x
hab : a + b = y'
k : K
hk : k • x = b
⊢ toLex ((truncLTLinearMap K c) (ofLex (↑↑f ⟨a, ha⟩))) +
      toLex ((truncLTLinearMap K c) ((RingHom.id K) k • ofLex (f.eval x))) ∈
    ((↑f).supSpanSingleton x (f.eval x) hx).toFun.range

Tactic:
map_smul,

Goal state after:
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
c : FiniteArchimedeanClass M
y' : M
hy'✝ : y' ∈ (f.extendFun hx).domain
hy' : y' ∈ (↑f).domain ⊔ K ∙ x
a : M
ha : a ∈ (↑f).domain
b : M
hb : b ∈ K ∙ x
hab : a + b = y'
k : K
hk : k • x = b
⊢ toLex ((truncLTLinearMap K c) (ofLex (↑↑f ⟨a, ha⟩))) +
      toLex ((RingHom.id K) k • (truncLTLinearMap K c) (ofLex (f.eval x))) ∈
    ((↑f).supSpanSingleton x (f.eval x) hx).toFun.range
Module: Mathlib.Algebra.Polynomial.Bivariate

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
x x✝ : R
⊢ eval₂ (evalRingHom x) x✝ = ⇑(eval₂RingHom (evalRingHom x) x✝)

Tactic:
coe_eval₂RingHom

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
x x✝ : R
⊢ eval₂ (evalRingHom x) x✝ = eval₂ (evalRingHom x) x✝
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case a
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
this : f = f.extend hx
⊢ False

Tactic:
have : f.val.domain = (f.extend hx).val.domain := by congr

Goal state after:
case a
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
this✝ : f = f.extend hx
this : (↑f).domain = (↑(f.extend hx)).domain
⊢ False
Module: Mathlib.Algebra.Polynomial.Bivariate

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
⊢ (MvPolynomial.aeval ![C X, Y]).comp (aevalAeval (MvPolynomial.X 0) (MvPolynomial.X 1)) = AlgHom.id R R[X][Y]

Tactic:
by ext <;> simp

Goal state after:
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
h : Nonempty (HahnEmbedding.Seed K M R)
f : ↥⊤ →ₗ[K] Lex R⟦FiniteArchimedeanClass M⟧
hpartial : HahnEmbedding.IsPartial h.some { domain := ⊤, toFun := f }
harch :
  ∀ (x : ↥(↑⟨{ domain := ⊤, toFun := f }, hpartial⟩).domain),
    (withTopOrderIso M) (ofLex (↑↑⟨{ domain := ⊤, toFun := f }, hpartial⟩ x)).orderTop = ArchimedeanClass.mk ↑x
⊢ StrictMono ⇑(f ∘ₗ LinearMap.codRestrict ⊤ LinearMap.id ⋯)

Tactic:
apply hpartial.strictMono.comp

Goal state after:
case refine_1
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
h : Nonempty (HahnEmbedding.Seed K M R)
f : ↥⊤ →ₗ[K] Lex R⟦FiniteArchimedeanClass M⟧
hpartial : HahnEmbedding.IsPartial h.some { domain := ⊤, toFun := f }
harch :
  ∀ (x : ↥(↑⟨{ domain := ⊤, toFun := f }, hpartial⟩).domain),
    (withTopOrderIso M) (ofLex (↑↑⟨{ domain := ⊤, toFun := f }, hpartial⟩ x)).orderTop = ArchimedeanClass.mk ↑x
⊢ StrictMono ⇑(LinearMap.codRestrict ⊤ LinearMap.id ⋯)
Module: Mathlib.Algebra.Polynomial.DenomsClearable

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
f : ℤ[X]
a b : ℤ
b0 : 0 < b
fab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0
⊢ ?m.59 * ↑b = 1

Tactic:
one_div_mul_cancel

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
f : ℤ[X]
a b : ℤ
b0 : 0 < b
fab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0
⊢ 1 = 1

K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
f : ℤ[X]
a b : ℤ
b0 : 0 < b
fab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0
⊢ ↑b ≠ 0
Module: Mathlib.Algebra.Polynomial.CoeffMem

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
p q : S[X]
Mp : Submodule R S
hp : ∀ (i : ℕ), p.coeff i ∈ Mp
hp' : 1 ∈ Mp
Mq : Submodule R S
hq : ∀ (i : ℕ), q.coeff i ∈ Mq
hq' : 1 ∈ Mq
i : ℕ
H : q.Monic
⊢ 1 ≤ Mq

Tactic:
by simpa

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
p q : S[X]
Mp : Submodule R S
hp : ∀ (i : ℕ), p.coeff i ∈ Mp
hp' : 1 ∈ Mp
Mq : Submodule R S
hq : ∀ (i : ℕ), q.coeff i ∈ Mq
hq' : 1 ∈ Mq
i : ℕ
H : q.Monic
⊢ 1 ≤ Mp

Tactic:
by simpa

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : R[X] ⊗[R] M →ₗ[R[X]] PolynomialModule R M := liftBaseChange R[X] (lsingle R 0)
inv : PolynomialModule R M →ₗ[R] R[X] ⊗[R] M := ↑R (eval X) ∘ₗ map R[X] ((TensorProduct.mk R R[X] M) 1)
left : inv ∘ₗ ↑R e = LinearMap.id
n : ℕ
x : M
a✝ : ℕ
⊢ (((↑R e ∘ₗ inv) ∘ₗ lsingle R n) x).coeff a✝ = ((LinearMap.id ∘ₗ lsingle R n) x).coeff a✝

Tactic:
simp [e, inv, ← monomial_one_right_eq_X_pow]

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

Goal state before:
pp: case isUnit_or_isUnit.inr.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror
h3 : IsRelPrime f f.mirror
g h : R[X]
fgh : f = g * h
k : R[X] := g * h.mirror
key : f * f.mirror = k * k.mirror
g_dvd_f : g ∣ f
h_dvd_f : h ∣ f
g_dvd_k : g ∣ k
h_dvd_k_rev : h ∣ k.mirror
hk : k = -f
⊢ IsUnit g ∨ IsUnit h

pp: case isUnit_or_isUnit.inr.inr.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror
h3 : IsRelPrime f f.mirror
g h : R[X]
fgh : f = g * h
k : R[X] := g * h.mirror
key : f * f.mirror = k * k.mirror
g_dvd_f : g ∣ f
h_dvd_f : h ∣ f
g_dvd_k : g ∣ k
h_dvd_k_rev : h ∣ k.mirror
hk : k = f.mirror
⊢ IsUnit g ∨ IsUnit h

pp: case isUnit_or_isUnit.inr.inr.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror
h3 : IsRelPrime f f.mirror
g h : R[X]
fgh : f = g * h
k : R[X] := g * h.mirror
key : f * f.mirror = k * k.mirror
g_dvd_f : g ∣ f
h_dvd_f : h ∣ f
g_dvd_k : g ∣ k
h_dvd_k_rev : h ∣ k.mirror
hk : k = -f.mirror
⊢ IsUnit g ∨ IsUnit h

Tactic:
· exact Or.inr (h3 h_dvd_f (by rwa [← neg_eq_iff_eq_neg.mpr hk, mirror_neg, dvd_neg]))

Goal state after:
case isUnit_or_isUnit.inr.inr.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror
h3 : IsRelPrime f f.mirror
g h : R[X]
fgh : f = g * h
k : R[X] := g * h.mirror
key : f * f.mirror = k * k.mirror
g_dvd_f : g ∣ f
h_dvd_f : h ∣ f
g_dvd_k : g ∣ k
h_dvd_k_rev : h ∣ k.mirror
hk : k = f.mirror
⊢ IsUnit g ∨ IsUnit h

case isUnit_or_isUnit.inr.inr.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror
h3 : IsRelPrime f f.mirror
g h : R[X]
fgh : f = g * h
k : R[X] := g * h.mirror
key : f * f.mirror = k * k.mirror
g_dvd_f : g ∣ f
h_dvd_f : h ∣ f
g_dvd_k : g ∣ k
h_dvd_k_rev : h ∣ k.mirror
hk : k = -f.mirror
⊢ IsUnit g ∨ IsUnit h
Module: Mathlib.Algebra.Polynomial.Homogenize

Goal state before:
pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (p.coeff k * (fun₀ | 0 => k | 1 => n - k).prod fun n e ↦ x n ^ e) = p.coeff k * (x 0 / x 1) ^ k * x 1 ^ n

Tactic:
Finsupp.update_eq_add_single,

Goal state after:
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (p.coeff k * ((fun₀ | 0 => k) + fun₀ | 1 => n - k).prod fun n e ↦ x n ^ e) = p.coeff k * (x 0 / x 1) ^ k * x 1 ^ n

case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (fun₀ | 0 => k) 1 = 0
Module: Mathlib.Algebra.Polynomial.Homogenize

Goal state before:
pp: case ha
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ≠ 0

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ k ≤ n

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ^ 0 = 1

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 0 ^ 0 = 1

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: case h_zero
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2), x a ^ 0 = 1

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: case h_add
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2) (b₁ b₂ : ℕ), x a ^ (b₁ + b₂) = x a ^ b₁ * x a ^ b₂

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (fun₀ | 0 => k) 1 = 0

Tactic:
simp_all [pow_add]

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
p : K[X]
hp2 : 2 ≤ p.natDegree
hp3 : p.natDegree ≤ 3
hp0 : p ≠ 0
⊢ Irreducible p ↔ p.roots = 0

Tactic:
rw [← irreducible_mul_leadingCoeff_inv,
      (monic_mul_leadingCoeff_inv hp0).irreducible_iff_roots_eq_zero_of_degree_le_three,
      mul_comm, roots_C_mul]

Goal state after:
case ha
K : Type u_1
inst✝ : Field K
p : K[X]
hp2 : 2 ≤ p.natDegree
hp3 : p.natDegree ≤ 3
hp0 : p ≠ 0
⊢ p.leadingCoeff⁻¹ ≠ 0

case hp2
K : Type u_1
inst✝ : Field K
p : K[X]
hp2 : 2 ≤ p.natDegree
hp3 : p.natDegree ≤ 3
hp0 : p ≠ 0
⊢ 2 ≤ (p * C p.leadingCoeff⁻¹).natDegree

case hp3
K : Type u_1
inst✝ : Field K
p : K[X]
hp2 : 2 ≤ p.natDegree
hp3 : p.natDegree ≤ 3
hp0 : p ≠ 0
⊢ (p * C p.leadingCoeff⁻¹).natDegree ≤ 3
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons.refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
⊢ ∀ i_1 ∈ Finset.cons i s hi,
    (Function.update (fun j ↦ r j * a j %ₘ g j) i ((q + ∑ j ∈ s, r j * b j %ₘ g i) %ₘ g i) i_1).degree < (g i_1).degree

Tactic:
rw [Finset.forall_mem_cons, Function.update_self]

Goal state after:
case cons.refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
⊢ ((q + ∑ j ∈ s, r j * b j %ₘ g i) %ₘ g i).degree < (g i).degree ∧
    ∀ x ∈ s,
      (Function.update (fun j ↦ r j * a j %ₘ g j) i ((q + ∑ j ∈ s, r j * b j %ₘ g i) %ₘ g i) x).degree < (g x).degree
Module: Mathlib.Algebra.Polynomial.UnitTrinomial

Goal state before:
pp: k m n : ℕ
hkm : k < m
hmn : m < n
x y z : ℤ
hp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)
hx : IsUnit ((C x * X ^ k).coeff k + (C y * X ^ m).coeff k + (C z * X ^ n).coeff k)
hy : IsUnit ((C x * X ^ k).coeff m + (C y * X ^ m).coeff m + (C z * X ^ n).coeff m)
hz : IsUnit ((C x * X ^ k).coeff n + (C y * X ^ m).coeff n + (C z * X ^ n).coeff n)
⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial

Tactic:
coeff_C_mul,

Goal state after:
k m n : ℕ
hkm : k < m
hmn : m < n
x y z : ℤ
hp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)
hx : IsUnit (x * (X ^ k).coeff k + y * (X ^ m).coeff k + z * (X ^ n).coeff k)
hy : IsUnit (x * (X ^ k).coeff m + y * (X ^ m).coeff m + z * (X ^ n).coeff m)
hz : IsUnit (x * (X ^ k).coeff n + y * (X ^ m).coeff n + z * (X ^ n).coeff n)
⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial
Module: Mathlib.Algebra.Polynomial.RuleOfSigns

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : ((X - C η) * P).natDegree = P.natDegree + 1
hndxeP : ((X - C η) * P.eraseLead).natDegree = P.natDegree
hQ : ((X - C η) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ P.coeff d - η * P.coeff (d + 1) < 0

Tactic:
sub_neg

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : ((X - C η) * P).natDegree = P.natDegree + 1
hndxeP : ((X - C η) * P.eraseLead).natDegree = P.natDegree
hQ : ((X - C η) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ P.coeff d < η * P.coeff (d + 1)
Module: Mathlib.Algebra.Polynomial.RuleOfSigns

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : ((X - C η) * P).natDegree = P.natDegree + 1
hndxeP : ((X - C η) * P.eraseLead).natDegree = P.natDegree
hQ : ((X - C η) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
hQ₁ : ((X - C η) * P).nextCoeff < 0
hndexP0 : ((X - C η) * P).eraseLead.natDegree = P.natDegree
this✝ : ((X - C η) * P).eraseLead.eraseLead = ((X - C η) * P.eraseLead).eraseLead
this :
  List.map (⇑SignType.sign) ((X - C η) * P).eraseLead.coeffList =
    List.map (⇑SignType.sign) ((X - C η) * P.eraseLead).coeffList
⊢ ((X - C η) * P).eraseLead.signVariations = ((X - C η) * P.eraseLead).signVariations

Tactic:
rw [signVariations, signVariations, this]

Goal state after: