mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: case inr.inl
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁷ : Semiring R
inst✝⁶ : LinearOrder B
inst✝⁵ : OrderBot B
p : R[A]
D : A → B
inst✝⁴ : AddZeroClass A
inst✝³ : Add B
inst✝² : AddLeftStrictMono B
inst✝¹ : AddRightStrictMono B
inst✝ : NoZeroDivisors R
hD : Function.Injective D
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
hp : p ≠ 0
⊢ leadingCoeff D (p * 0) = leadingCoeff D p * leadingCoeff D 0

Tactic:
[

Goal state after:
case inr.inl
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁷ : Semiring R
inst✝⁶ : LinearOrder B
inst✝⁵ : OrderBot B
p : R[A]
D : A → B
inst✝⁴ : AddZeroClass A
inst✝³ : Add B
inst✝² : AddLeftStrictMono B
inst✝¹ : AddRightStrictMono B
inst✝ : NoZeroDivisors R
hD : Function.Injective D
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
hp : p ≠ 0
⊢ leadingCoeff D (p * 0) = leadingCoeff D p * leadingCoeff D 0
Module: Mathlib.Algebra.Polynomial.Eval.Coeff

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
hf : Function.Injective ⇑f
⊢ (map f p).support = p.support

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
hf : Function.Injective ⇑f
⊢ (map f p).support = p.support
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: case inr
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : LinearOrder B
inst✝⁴ : OrderBot B
p q : R[A]
D : A → B
inst✝³ : AddZeroClass A
inst✝² : Add B
inst✝¹ : AddLeftStrictMono B
inst✝ : AddRightStrictMono B
hD : Function.Injective D
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
hq : Monic D q
hp : p ≠ 0
⊢ leadingCoeff D (p * q) = leadingCoeff D p

Tactic:
rw [leadingCoeff, hq.supDegree_mul_of_ne_zero_left hD hadd hp,
    coeff_supDegree_add_supDegree hD hadd, hq, mul_one]

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

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
inst✝ : Nontrivial R
i j : ℕ
⊢ (monomial i) 1 = (monomial j) 1 ↔ i = j

Tactic:
[

Goal state after:
R : Type u
inst✝¹ : Semiring R
inst✝ : Nontrivial R
i j : ℕ
⊢ (monomial i) 1 = (monomial j) 1 ↔ i = j
Module: Mathlib.Algebra.Polynomial.Eval.Degree

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

Tactic:
[

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
r : R
n : ℕ
⊢ (p.comp (C r * X)).coeff n = p.coeff n * r ^ n
Module: Mathlib.Algebra.Polynomial.Eval.Degree

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
r : R
n : ℕ
⊢ (∑ x ∈ range (p.natDegree + 1), C (p.coeff x) * (C r ^ x * X ^ x)).coeff n = p.coeff n * r ^ n

Tactic:
← C_pow,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
r : R
n : ℕ
⊢ (∑ x ∈ range (p.natDegree + 1), C (p.coeff x) * (C (r ^ x) * X ^ x)).coeff n = p.coeff n * r ^ n
Module: Mathlib.Algebra.Polynomial.Eval.Degree

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
r : R
n b : ℕ
a✝ : b ∈ range (p.natDegree + 1)
h : b ≠ n
⊢ p.coeff b * (r ^ b * if n = b then 1 else 0) = 0

Tactic:
[

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
r : R
n b : ℕ
a✝ : b ∈ range (p.natDegree + 1)
h : b ≠ n
⊢ p.coeff b * (r ^ b * if n = b then 1 else 0) = 0
Module: Mathlib.Algebra.Polynomial.Eval.Degree

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
r : R
hr : r ∈ nonZeroDivisors R
⊢ p.comp (C r * X) = 0 ↔ p = 0

Tactic:
[

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
r : R
hr : r ∈ nonZeroDivisors R
⊢ p.comp (C r * X) = 0 ↔ p = 0
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case mp
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
h : x + y = single a n
⊢ single a (x a) = x ∧ single a (y a) = y

Tactic:
[

Goal state after:
case mp
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
h : x + y = single a n
⊢ single a (x a) = x ∧ single a (y a) = y
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case mp
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
i : α
h : (x + y) i = (single a n) i
⊢ (single a (x a)) i = x i ∧ (single a (y a)) i = y i

Tactic:
[

Goal state after:
case mp
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
i : α
h : (x + y) i = (single a n) i
⊢ (single a (x a)) i = x i ∧ (single a (y a)) i = y i
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case mp.inr
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
i : α
h : x i + y i = if a = i then n else 0
hai : a ≠ i
⊢ (if a = i then x a else 0) = x i ∧ (if a = i then y a else 0) = y i

Tactic:
[

Goal state after:
case mp.inr
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
i : α
h : x i + y i = if a = i then n else 0
hai : a ≠ i
⊢ (if a = i then x a else 0) = x i ∧ (if a = i then y a else 0) = y i
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case refine_3
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
f : α ↪ β
y : α →₀ ℕ
u v : β →₀ ℕ
h : u + v = embDomain f y
⊢ embDomain f (comapDomain (⇑f) u ⋯) = u

Tactic:
embDomain_comapDomain ((mem_range_embDomain_iff ..).mp
      (isLowerSet_range_embDomain f (le_iff_exists_add.mpr ⟨v, h.symm⟩) (by simp)))

Goal state after:
case refine_3
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
f : α ↪ β
y : α →₀ ℕ
u v : β →₀ ℕ
h : u + v = embDomain f y
⊢ u = u
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case refine_2
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
x : α →₀ ℕ
y : β →₀ ℕ
u v : α ⊕ β →₀ ℕ
h : u + v = x.sumElim y
⊢ comapDomain Sum.inl u ⋯ + comapDomain Sum.inl v ⋯ = x

Tactic:
rw [← comapDomain_add_of_injective Sum.inl_injective, h, comapDomain_inl_sumElim]

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

Goal state before:
pp: R : Type u
S : Type v
T : Type w
A : Type z
A' : Type u_1
B : Type u_2
a b : R
n : ℕ
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
p✝ q r✝ : R[X]
r : R
p : A[X]
⊢ (C ((algebraMap R A) r) * p).toFinsupp = (p * C ((algebraMap R A) r)).toFinsupp

Tactic:
[

Goal state after:
R : Type u
S : Type v
T : Type w
A : Type z
A' : Type u_1
B : Type u_2
a b : R
n : ℕ
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
p✝ q r✝ : R[X]
r : R
p : A[X]
⊢ (C ((algebraMap R A) r) * p).toFinsupp = (p * C ((algebraMap R A) r)).toFinsupp
Module: Mathlib.Data.Finsupp.Order

Goal state before:
pp: α : Type u_5
M : Type u_6
N : Type u_7
inst✝³ : Zero M
inst✝² : AddCommMonoid N
inst✝¹ : PartialOrder N
inst✝ : IsOrderedAddMonoid N
f : α →₀ M
g : M → N
hg : g 0 = 0
h : 0 ≤ fun x ↦ g x
a : α
⊢ g (f a) ≤ f.sum fun x m ↦ g m

Tactic:
simp only [← sum_single_index (h := fun (_ : α) m ↦ g m) (a := a) (b := f a) hg]

Goal state after:
α : Type u_5
M : Type u_6
N : Type u_7
inst✝³ : Zero M
inst✝² : AddCommMonoid N
inst✝¹ : PartialOrder N
inst✝ : IsOrderedAddMonoid N
f : α →₀ M
g : M → N
hg : g 0 = 0
h : 0 ≤ fun x ↦ g x
a : α
⊢ ((single a (f a)).sum fun x m ↦ g m) ≤ f.sum fun x m ↦ g m
Module: Mathlib.Data.Finsupp.Order

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
f : ι ↪ κ
f1 f2 : ι →₀ α
⊢ embDomain f (f1 - f2) = embDomain f f1 - embDomain f f2

Tactic:
[

Goal state after:
ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
f : ι ↪ κ
f1 f2 : ι →₀ α
⊢ embDomain f (f1 - f2) = embDomain f f1 - embDomain f f2
Module: Mathlib.Order.PiLex

Goal state before:
pp: ι : Type u_1
β : ι → Type u_2
inst✝ : (i : ι) → PartialOrder (β i)
r : ι → ι → Prop
hwf : WellFounded r
x y : (i : ι) → β i
hlt : x < y
⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) x y

Tactic:
[

Goal state after:
ι : Type u_1
β : ι → Type u_2
inst✝ : (i : ι) → PartialOrder (β i)
r : ι → ι → Prop
hwf : WellFounded r
x y : (i : ι) → β i
hlt : x < y
⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) x y
Module: Mathlib.Order.PiLex

Goal state before:
pp: ι : Type u_1
β : ι → Type u_2
inst✝² : Unique ι
inst✝¹ : LinearOrder ι
inst✝ : (i : ι) → PartialOrder (β i)
x y : Lex ((i : ι) → β i)
⊢ x ≤ y ↔ x default ≤ y default

Tactic:
[

Goal state after:
ι : Type u_1
β : ι → Type u_2
inst✝² : Unique ι
inst✝¹ : LinearOrder ι
inst✝ : (i : ι) → PartialOrder (β i)
x y : Lex ((i : ι) → β i)
⊢ x ≤ y ↔ x default ≤ y default
Module: Mathlib.Order.PiLex

Goal state before:
pp: ι : Type u_1
β : ι → Type u_2
inst✝² : Unique ι
inst✝¹ : LinearOrder ι
inst✝ : (i : ι) → PartialOrder (β i)
x y : Colex ((i : ι) → β i)
⊢ x ≤ y ↔ x default ≤ y default

Tactic:
[

Goal state after:
ι : Type u_1
β : ι → Type u_2
inst✝² : Unique ι
inst✝¹ : LinearOrder ι
inst✝ : (i : ι) → PartialOrder (β i)
x y : Colex ((i : ι) → β i)
⊢ x ≤ y ↔ x default ≤ y default
Module: Mathlib.Order.PiLex

Goal state before:
pp: ι : Type u_1
β : ι → Type u_2
inst✝² : LinearOrder ι
x : (i : ι) → β i
i : ι
a : β i
inst✝¹ : (i : ι) → PartialOrder (β i)
inst✝ : WellFoundedLT ι
⊢ toLex x ≤ toLex (update x i a) ↔ x i ≤ a

Tactic:
[

Goal state after:
ι : Type u_1
β : ι → Type u_2
inst✝² : LinearOrder ι
x : (i : ι) → β i
i : ι
a : β i
inst✝¹ : (i : ι) → PartialOrder (β i)
inst✝ : WellFoundedLT ι
⊢ toLex x ≤ toLex (update x i a) ↔ x i ≤ a
Module: Mathlib.Order.PiLex

Goal state before:
pp: ι : Type u_1
β : ι → Type u_2
inst✝² : LinearOrder ι
x : (i : ι) → β i
i : ι
a : β i
inst✝¹ : (i : ι) → PartialOrder (β i)
inst✝ : WellFoundedLT ι
⊢ toLex (update x i a) ≤ toLex x ↔ a ≤ x i

Tactic:
[

Goal state after:
ι : Type u_1
β : ι → Type u_2
inst✝² : LinearOrder ι
x : (i : ι) → β i
i : ι
a : β i
inst✝¹ : (i : ι) → PartialOrder (β i)
inst✝ : WellFoundedLT ι
⊢ toLex (update x i a) ≤ toLex x ↔ a ≤ x i
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
p : R[X]
x : S
⊢ (aeval x) p = ∑ i ∈ range (p.natDegree + 1), p.coeff i • x ^ i

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
p : R[X]
x : S
⊢ (aeval x) p = ∑ i ∈ range (p.natDegree + 1), p.coeff i • x ^ i
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
p : R[X]
n : ℕ
hn : p.natDegree < n
x : S
⊢ (aeval x) p = ∑ i ∈ range n, p.coeff i • x ^ i

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
p : R[X]
n : ℕ
hn : p.natDegree < n
x : S
⊢ (aeval x) p = ∑ i ∈ range n, p.coeff i • x ^ i
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: case add
R : Type u
A : Type z
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
q : Submodule R M
m : M
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
hm : m ∈ q
a : A
hq : q ≤ Submodule.comap ((Algebra.lsmul R R M) a) q
f₁ f₂ : R[X]
h₁ : (aeval a) f₁ • m ∈ q
h₂ : (aeval a) f₂ • m ∈ q
⊢ (aeval a) (f₁ + f₂) • m ∈ q

Tactic:
[

Goal state after:
case add
R : Type u
A : Type z
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
q : Submodule R M
m : M
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
hm : m ∈ q
a : A
hq : q ≤ Submodule.comap ((Algebra.lsmul R R M) a) q
f₁ f₂ : R[X]
h₁ : (aeval a) f₁ • m ∈ q
h₂ : (aeval a) f₂ • m ∈ q
⊢ (aeval a) (f₁ + f₂) • m ∈ q
Module: Mathlib.Data.DFinsupp.NeLocus

Goal state before:
pp: α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → Zero (N a)
f g : Π₀ (a : α), N a
⊢ f.neLocus g = g.neLocus f

Tactic:
[

Goal state after:
α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → Zero (N a)
f g : Π₀ (a : α), N a
⊢ f.neLocus g = g.neLocus f
Module: Mathlib.Data.DFinsupp.NeLocus

Goal state before:
pp: α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → AddGroup (N a)
f g : Π₀ (a : α), N a
⊢ (f + g).neLocus f = g.support

Tactic:
by
  rw [neLocus_comm, neLocus_self_add_right]

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

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
α : Type u_2
s : Finset α
f : α → σ →₀ ℕ
g : α → R
⊢ (monomial (∑ i ∈ s, f i)) (∏ i ∈ s, g i) = ∏ i ∈ s, (monomial (f i)) (g i)

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
α : Type u_2
s : Finset α
f : α → σ →₀ ℕ
g : α → R
⊢ (monomial (∑ i ∈ s, f i)) (∏ i ∈ s, g i) = ∏ i ∈ s, (monomial (f i)) (g i)
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
d : σ →₀ ℕ
x : σ → ℕ
s : Finset σ
⊢ coeff d (∏ y ∈ s, X y ^ x y) = if d = Finsupp.indicator s fun i x_1 ↦ x i then 1 else 0

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
d : σ →₀ ℕ
x : σ → ℕ
s : Finset σ
⊢ coeff d (∏ y ∈ s, X y ^ x y) = if d = Finsupp.indicator s fun i x_1 ↦ x i then 1 else 0
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: case pos
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
m s : σ →₀ ℕ
r : R
p : MvPolynomial σ R
h : s ≤ m
⊢ coeff m (p * (monomial s) r) = coeff (m - s + s) (p * (monomial s) r)

Tactic:
rw [tsub_add_cancel_of_le h]

Goal state after:
Module: Mathlib.Data.DFinsupp.Lex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → PartialOrder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt : x < y
⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) ⇑x ⇑y

Tactic:
[

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → PartialOrder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt : x < y
⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) ⇑x ⇑y
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
m : σ →₀ ℕ
s : σ
p : MvPolynomial σ R
⊢ (if Finsupp.single s 1 ≤ m then coeff (m - Finsupp.single s 1) p * 1 else 0) =
    if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
m : σ →₀ ℕ
s : σ
p : MvPolynomial σ R
⊢ (if Finsupp.single s 1 ≤ m then coeff (m - Finsupp.single s 1) p * 1 else 0) =
    if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
m : σ →₀ ℕ
s : σ
p : MvPolynomial σ R
⊢ (if Finsupp.single s 1 ≤ m then 1 * coeff (m - Finsupp.single s 1) p else 0) =
    if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
m : σ →₀ ℕ
s : σ
p : MvPolynomial σ R
⊢ (if Finsupp.single s 1 ≤ m then 1 * coeff (m - Finsupp.single s 1) p else 0) =
    if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p q : MvPolynomial σ R
⊢ (p + q).degrees ≤ p.degrees ⊔ q.degrees

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p q : MvPolynomial σ R
⊢ (p + q).degrees ≤ p.degrees ⊔ q.degrees
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: case C
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
r : R
m : σ →₀ ℕ
⊢ coeff m ((map f) (C r)) = f (coeff m (C r))

Tactic:
[

Goal state after:
case C
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
r : R
m : σ →₀ ℕ
⊢ coeff m ((map f) (C r)) = f (coeff m (C r))
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
ι : Type u_3
inst✝ : DecidableEq σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ (∑ i ∈ s, f i).degrees ≤ s.sup fun i ↦ (f i).degrees

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
ι : Type u_3
inst✝ : DecidableEq σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ (∑ i ∈ s, f i).degrees ≤ s.sup fun i ↦ (f i).degrees
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
⊢ (p * q).degrees ≤ p.degrees + q.degrees

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
⊢ (p * q).degrees ≤ p.degrees + q.degrees
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
⊢ p.degrees = 0 ↔ p.support ⊆ {0}

Tactic:
rw [Finset.subset_singleton_iff', Multiset.eq_zero_iff_forall_notMem]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
⊢ (∀ (a : σ), a ∉ p.degrees) ↔ ∀ b ∈ p.support, b = 0
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: case refine_1
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : ∀ (a : σ), a ∉ p.degrees
s : σ →₀ ℕ
hs : s ∈ p.support
⊢ s.support = ∅

Tactic:
simp only [mem_degrees] at h

Goal state after:
case refine_1
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
s : σ →₀ ℕ
hs : s ∈ p.support
h : ∀ (a : σ), ¬∃ d, coeff d p ≠ 0 ∧ a ∈ d.support
⊢ s.support = ∅
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
g : S₁ → R
hg : g 0 = 0
φ : MvPolynomial σ S₁
⊢ (map f) (ofCoeff (mapRange g hg φ.coeff)) = φ ↔ ∀ (d : σ →₀ ℕ), f (g (coeff d φ)) = coeff d φ

Tactic:
[

Goal state after:
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
g : S₁ → R
hg : g 0 = 0
φ : MvPolynomial σ S₁
⊢ (map f) (ofCoeff (mapRange g hg φ.coeff)) = φ ↔ ∀ (d : σ →₀ ℕ), f (g (coeff d φ)) = coeff d φ
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f g : MvPolynomial σ R
⊢ degreeOf n (f + g) ≤ max (degreeOf n f) (degreeOf n g)

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f g : MvPolynomial σ R
⊢ degreeOf n (f + g) ≤ max (degreeOf n f) (degreeOf n g)
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
φ ψ : MvPolynomial σ R
⊢ (φ * ψ).vars ⊆ φ.vars ∪ ψ.vars

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
φ ψ : MvPolynomial σ R
⊢ (φ * ψ).vars ⊆ φ.vars ∪ ψ.vars
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: case neg
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
a : S₁
h : ¬a = 0
hx : a ∈ range ⇑f
⊢ C a ∈ range ⇑(map f)

Tactic:
obtain ⟨b, rfl⟩ := hx

Goal state after:
case neg
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
b : R
h : ¬f b = 0
⊢ C (f b) ∈ range ⇑(map f)
Module: Mathlib.Algebra.MvPolynomial.CommRing

Goal state before:
pp: case inr.inl
R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p : MvPolynomial σ R
inst✝ : DecidableEq σ
d d' : σ →₀ ℕ
c : R
hdd' : d ≠ d'
hc : coeff d p = c
x : σ →₀ ℕ
hx : x ∈ (p - ((monomial d) c - (monomial d') c)).support
hd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support
hdelta : x ∈ ((monomial d) c - (monomial d') c).support
h1 : x ∈ ((monomial d) c).support
⊢ x ∈ p.support.erase d ∪ {d'}

Tactic:
support_monomial

Goal state after:
case inr.inl
R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p : MvPolynomial σ R
inst✝ : DecidableEq σ
d d' : σ →₀ ℕ
c : R
hdd' : d ≠ d'
hc : coeff d p = c
x : σ →₀ ℕ
hx : x ∈ (p - ((monomial d) c - (monomial d') c)).support
hd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support
hdelta : x ∈ ((monomial d) c - (monomial d') c).support
h1 : x ∈ if c = 0 then ∅ else {d}
⊢ x ∈ p.support.erase d ∪ {d'}
Module: Mathlib.Algebra.MvPolynomial.CommRing

Goal state before:
pp: case inr.inr
R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p : MvPolynomial σ R
inst✝ : DecidableEq σ
d d' : σ →₀ ℕ
c : R
hdd' : d ≠ d'
hc : coeff d p = c
x : σ →₀ ℕ
hx : x ∈ (p - ((monomial d) c - (monomial d') c)).support
hd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support
hdelta : x ∈ ((monomial d) c - (monomial d') c).support
h2 : x ∈ ((monomial d') c).support
⊢ x ∈ p.support.erase d ∪ {d'}

Tactic:
support_monomial

Goal state after:
case inr.inr
R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p : MvPolynomial σ R
inst✝ : DecidableEq σ
d d' : σ →₀ ℕ
c : R
hdd' : d ≠ d'
hc : coeff d p = c
x : σ →₀ ℕ
hx : x ∈ (p - ((monomial d) c - (monomial d') c)).support
hd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support
hdelta : x ∈ ((monomial d) c - (monomial d') c).support
h2 : x ∈ if c = 0 then ∅ else {d'}
⊢ x ∈ p.support.erase d ∪ {d'}
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
ι : Type u_3
i : σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ degreeOf i (∑ j ∈ s, f j) ≤ s.sup fun j ↦ degreeOf i (f j)

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
ι : Type u_3
i : σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ degreeOf i (∑ j ∈ s, f j) ≤ s.sup fun j ↦ degreeOf i (f j)
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
ι : Type u_3
i : σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, degreeOf i (f j)

Tactic:
[

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
ι : Type u_3
i : σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, degreeOf i (f j)
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
ι : Type u_3
i : σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, degreeOf i (f j)

Tactic:
simp_rw [degreeOf_eq_sup]
  exact supDegree_prod_le (by simp only [coe_zero, Pi.zero_apply]) (by simp)

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S₁
inst✝² : CommSemiring S₂
inst✝¹ : Algebra R S₁
inst✝ : Algebra R S₂
f : S₁ →ₐ[R] S₂
x✝ : MvPolynomial σ S₂
⊢ x✝ ∈ range ⇑(mapAlgHom f) ↔ x✝ ∈ ↑(coeffsIn σ (Subalgebra.toSubmodule f.range))

Tactic:
mem_range_map_iff_coeffs_subset,

Goal state after:
R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S₁
inst✝² : CommSemiring S₂
inst✝¹ : Algebra R S₁
inst✝ : Algebra R S₂
f : S₁ →ₐ[R] S₂
x✝ : MvPolynomial σ S₂
⊢ ↑x✝.coeffs ⊆ range ⇑↑f ↔ x✝ ∈ ↑(coeffsIn σ (Subalgebra.toSubmodule f.range))
Module: Mathlib.Data.Finsupp.Fin

Goal state before:
pp: n : ℕ
M : Type u_1
inst✝ : Zero M
y : M
t : Fin (n + 1) →₀ M
⊢ (t.update 0 y).tail = t.tail

Tactic:
simp [tail]

Goal state after:
Module: Mathlib.Data.Finsupp.Fin

Goal state before:
pp: n : ℕ
i : Fin n
M : Type u_1
inst✝ : Zero M
y : M
t : Fin (n + 1) →₀ M
a✝ : Fin n
⊢ (t.update i.succ y).tail a✝ = (t.tail.update i y) a✝

Tactic:
simp [tail]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Finsupp.Fin

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Zero M
inst✝ : AddCommMonoid N
n : ℕ
σ : Fin n →₀ M
i : M
f : Fin (n + 1) → M → N
h : ∀ (x : Fin (n + 1)), f x 0 = 0
⊢ ∑ i_1, f i_1 ((cons i σ) i_1) = f 0 i + ∑ i, Fin.tail f i (σ i)

Tactic:
[

Goal state after:
M : Type u_1
N : Type u_2
inst✝¹ : Zero M
inst✝ : AddCommMonoid N
n : ℕ
σ : Fin n →₀ M
i : M
f : Fin (n + 1) → M → N
h : ∀ (x : Fin (n + 1)), f x 0 = 0
⊢ ∑ i_1, f i_1 ((cons i σ) i_1) = f 0 i + ∑ i, Fin.tail f i (σ i)