mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: case inr
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ a ∈ p.degrees, ∃ b ∈ q.degrees, a = b

Tactic:
]

Goal state after:
case inr
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ a ∈ p.degrees, ∃ b ∈ q.degrees, a = b
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 : σ → R
p : MvPolynomial σ R
⊢ eval₂ f (⇑f ∘ g) ((map (RingHom.id R)) p) = eval₂ f (⇑f ∘ g) ((map (RingHom.id R)) p)

Tactic:
]

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

Goal state before:
pp: case inr.refine_1
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ d, coeff d p ≠ 0 ∧ j ∈ d.support

Tactic:
]

Goal state after:
case inr.refine_1
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ d, coeff d p ≠ 0 ∧ j ∈ d.support
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
inst✝ : CommSemiring S₂
f : R →+* S₁
g : σ → S₂
φ : S₁ →+* S₂
p : MvPolynomial σ R
⊢ (eval g) ((map (φ.comp f)) p) = (eval g) ((map (φ.comp f)) p)

Tactic:
]

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

Goal state before:
pp: case inr.refine_2
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ d, coeff d q ≠ 0 ∧ j ∈ d.support

Tactic:
]

Goal state after:
case inr.refine_2
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
d : σ →₀ ℕ
hd : coeff d p ≠ 0
j : σ
hj : j ∈ d.support
h : coeff d q ≠ 0
⊢ ∃ d, coeff d q ≠ 0 ∧ j ∈ d.support
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: case refine_1.mem_mul_mem
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
r : S
s : σ →₀ ℕ
m : S
hm : m ∈ M
n : S
hn : n ∈ N
⊢ (monomial s) m * (monomial 0) n ∈ coeffsIn σ M * coeffsIn σ N

Tactic:
]

Goal state after:
case refine_1.mem_mul_mem
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
r : S
s : σ →₀ ℕ
m : S
hm : m ∈ M
n : S
hn : n ∈ N
⊢ (monomial s) m * (monomial 0) n ∈ coeffsIn σ M * coeffsIn σ N
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
p : MvPolynomial σ R
x : σ →₀ ℕ
hx : coeff x p = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
s : Finset σ
q : MvPolynomial (↥s) R
n : ℕ := Fintype.card ↥s
e : ↥s ≃ Fin (Fintype.card ↥s) := Fintype.equivFin ↥s
⊢ (rename Subtype.val) q = (rename Subtype.val) ((rename (⇑e.symm ∘ ⇑e)) q)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
s : Finset σ
q : MvPolynomial (↥s) R
n : ℕ := Fintype.card ↥s
e : ↥s ≃ Fin (Fintype.card ↥s) := Fintype.equivFin ↥s
⊢ (rename Subtype.val) q = (rename Subtype.val) ((rename (⇑e.symm ∘ ⇑e)) q)
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: case refine_2
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
⊢ ∀ m ∈ coeffsIn σ M, ∀ n ∈ coeffsIn σ N, m * n ∈ coeffsIn σ (M * N)

Tactic:
]

Goal state after:
case refine_2
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
⊢ ∀ m ∈ coeffsIn σ M, ∀ n ∈ coeffsIn σ N, m * n ∈ coeffsIn σ (M * N)
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: case H₂
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p : MvPolynomial σ R
f : R →+* S₁
hf : Injective ⇑f
x : σ →₀ ℕ
hx : f (coeff x p) = f 0
⊢ coeff x p = 0

Tactic:
]

Goal state after:
case H₂
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p : MvPolynomial σ R
f : R →+* S₁
hf : Injective ⇑f
x : σ →₀ ℕ
hx : f (coeff x p) = f 0
⊢ coeff x p = 0
Module: Mathlib.Algebra.MvPolynomial.Rename

Goal state before:
pp: case monomial
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
hf : Injective f
d u : σ →₀ ℕ
r : R
⊢ (if Finsupp.mapDomain f u = Finsupp.mapDomain f d then r else 0) = if u = d then r else 0

Tactic:
]

Goal state after:
case monomial
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
hf : Injective f
d u : σ →₀ ℕ
r : R
⊢ (if Finsupp.mapDomain f u = Finsupp.mapDomain f d then r else 0) = if u = d then r else 0
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
q : R →+* S₁
r : R
hr : ∀ (r' : R), q r' = 0 ↔ r ∣ r'
φ : MvPolynomial σ R
⊢ (∀ (i : σ →₀ ℕ), r ∣ coeff i φ) ↔ ∀ (m : σ →₀ ℕ), coeff m ((map q) φ) = coeff m 0

Tactic:
]

Goal state after:
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
q : R →+* S₁
r : R
hr : ∀ (r' : R), q r' = 0 ↔ r ∣ r'
φ : MvPolynomial σ R
⊢ (∀ (i : σ →₀ ℕ), r ∣ coeff i φ) ↔ ∀ (m : σ →₀ ℕ), coeff m ((map q) φ) = coeff m 0
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: case refine_2
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
x : MvPolynomial σ S
hx : x ∈ coeffsIn σ M
y : MvPolynomial σ S
hy : y ∈ coeffsIn σ N
k : σ →₀ ℕ
⊢ ∑ x_1 ∈ Finset.antidiagonal k, coeff x_1.1 x * coeff x_1.2 y ∈ M * N

Tactic:
]

Goal state after:
case refine_2
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M N : Submodule R S
x : MvPolynomial σ S
hx : x ∈ coeffsIn σ M
y : MvPolynomial σ S
hy : y ∈ coeffsIn σ N
k : σ →₀ ℕ
⊢ ∑ x_1 ∈ Finset.antidiagonal k, coeff x_1.1 x * coeff x_1.2 y ∈ M * N
Module: Mathlib.Algebra.MvPolynomial.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ ↪ τ
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ coeff d φ = coeff d φ

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ d ∉ ((rename f) φ).support

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ d ∉ ((rename f) φ).support
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
⊢ (∃ d, coeff d ((rename f) φ) ≠ 0 ∧ i ∈ d.support) → ∃ a ∈ φ.degrees, f a = i

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
⊢ (∃ d, coeff d ((rename f) φ) ≠ 0 ∧ i ∈ d.support) → ∃ a ∈ φ.degrees, f a = i
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
n : ℕ
x✝ : n + 2 ≠ 0
M : Submodule R S
⊢ coeffsIn σ (M ^ (n + 1) * M) = coeffsIn σ M ^ (n + 2)

Tactic:
coeffsIn_mul,

Goal state after:
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
n : ℕ
x✝ : n + 2 ≠ 0
M : Submodule R S
⊢ coeffsIn σ (M ^ (n + 1)) * coeffsIn σ M = coeffsIn σ M ^ (n + 2)
Module: Mathlib.Algebra.MvPolynomial.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
H : ∃ a ∈ φ.coeff.support, (mapDomain.addMonoidHom f) a = d
⊢ False

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
H : ∃ a ∈ φ.coeff.support, (mapDomain.addMonoidHom f) a = d
⊢ False
Module: Mathlib.Algebra.MvPolynomial.Basic

Goal state before:
pp: R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
n : ℕ
x✝ : n + 2 ≠ 0
M : Submodule R S
⊢ coeffsIn σ M ^ (n + 1 + 1) = coeffsIn σ M ^ (n + 2)

pp: case a
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
n : ℕ
x✝ : n + 2 ≠ 0
M : Submodule R S
⊢ n + 1 ≠ 0

Tactic:
]

Goal state after:
case a
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
n : ℕ
x✝ : n + 2 ≠ 0
M : Submodule R S
⊢ n + 1 ≠ 0
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : i ∈ (Finsupp.mapDomain f x).support
⊢ ∃ a ∈ φ.degrees, f a = i

Tactic:
simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi

Goal state after:
R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : (x.sum fun a ↦ Finsupp.single (f a)) i ≠ 0
⊢ ∃ a ∈ φ.degrees, f a = i
Module: Mathlib.Algebra.MvPolynomial.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
hu : φ.coeff u ≠ 0
h : coeff u φ = 0
⊢ False

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
hu : φ.coeff u ≠ 0
h : coeff u φ = 0
⊢ False
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : ∑ a ∈ x.support, (Finsupp.single (f a) (x a)) i ≠ 0
⊢ ∃ a ∈ φ.degrees, f a = i

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : ∑ a ∈ x.support, (Finsupp.single (f a) (x a)) i ≠ 0
⊢ ∃ a ∈ φ.degrees, f a = i
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ ((map f) ((monomial a) s)).coeffs ∪ ((map f) p).coeffs ⊆ Finset.image (⇑f) (((monomial a) s).coeffs ∪ p.coeffs)

pp: case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ Disjoint ((map f) ((monomial a) s)).support ((map f) p).support

Tactic:
]

Goal state after:
case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ ((map f) ((monomial a) s)).coeffs ∪ ((map f) p).coeffs ⊆ Finset.image (⇑f) (((monomial a) s).coeffs ∪ p.coeffs)

case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ Disjoint ((map f) ((monomial a) s)).support ((map f) p).support
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : ∀ a ∈ φ.degrees, f a ≠ i
⊢ 0 = 0

pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : ∀ a ∈ φ.degrees, f a ≠ i
⊢ ∀ x_1 ∈ x.support, (Finsupp.single (f x_1) (x x_1)) i = 0

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
hi : ∀ a ∈ φ.degrees, f a ≠ i
⊢ ∀ x_1 ∈ x.support, (Finsupp.single (f x_1) (x x_1)) i = 0
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p : MvPolynomial σ R
⊢ p.degrees.toFinset = p.degrees.toFinset

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p : MvPolynomial σ R
⊢ p.degrees.toFinset = p.degrees.toFinset
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
r : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
h : r ≠ 0
⊢ s.support = s.support

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
r : R
inst✝ : CommSemiring R
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
n : σ
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
⊢ {n} = {n}

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
i : τ
x : σ →₀ ℕ
hx : coeff x φ ≠ 0
hd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0
j : σ
hj : j ∈ x.support
hi : f j ≠ i
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs ∪ Finset.image (⇑f) p.coeffs ∧
    ((map f) p).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs ∪ Finset.image (⇑f) p.coeffs

Tactic:
]

Goal state after:
case monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
f : R →+* S₁
inst✝ : DecidableEq S₁
a : σ →₀ ℕ
s : R
p : MvPolynomial σ R
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
ih : ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs
⊢ ((map f) ((monomial a) s)).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs ∪ Finset.image (⇑f) p.coeffs ∧
    ((map f) p).coeffs ⊆ Finset.image (⇑f) ((monomial a) s).coeffs ∪ Finset.image (⇑f) p.coeffs
Module: Mathlib.Algebra.MvPolynomial.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
p : MvPolynomial τ R
f : σ → τ
hf : Injective f
⊢ {x ∈ p.support | x ∈ range (Finsupp.mapDomain f)} ⊆ p.support

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
p : MvPolynomial τ R
f : σ → τ
hf : Injective f
⊢ {x ∈ p.support | x ∈ range (Finsupp.mapDomain f)} ⊆ p.support
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p = C (coeff 0 p)
⊢ (C (coeff 0 p)).vars = ∅

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p = C (coeff 0 p)
⊢ (C (coeff 0 p)).vars = ∅
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
n : σ
p : MvPolynomial σ R
⊢ Multiset.count n p.degrees = Multiset.count n p.degrees

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
n : σ
p : MvPolynomial σ R
⊢ Multiset.count n p.degrees = Multiset.count n p.degrees
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
⊢ (f.support.sup fun a ↦ Multiset.count n (toMultiset a)) = f.support.sup fun m ↦ m n

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
⊢ (f.support.sup fun a ↦ Multiset.count n (toMultiset a)) = f.support.sup fun m ↦ m n
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p.vars = ∅
⊢ p.totalDegree = 0

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p.vars = ∅
⊢ p.totalDegree = 0
Module: Mathlib.Algebra.MvPolynomial.CommRing

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommRing R
p : MvPolynomial σ R
d d' : σ →₀ ℕ
c : R
hdd' : d ≠ d'
hc : coeff d p = c
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p.degrees = 0
⊢ p.degrees.card = 0

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : p.degrees = 0
⊢ p.degrees.card = 0
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
d : ℕ
h : 0 < d
⊢ (∀ b ∈ f.support, b n < d) ↔ ∀ m ∈ f.support, m n < d

pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
d : ℕ
h : 0 < d
⊢ ⊥ < d

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
d : ℕ
h : 0 < d
⊢ ⊥ < d
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
n : σ
f : MvPolynomial σ R
d : ℕ
⊢ (∀ b ∈ f.support, b n ≤ d) ↔ ∀ m ∈ f.support, m n ≤ d

Tactic:
]

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

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : DecidableEq σ
x : σ
h : Disjoint p.degrees q.degrees
hx : x ∈ p.degrees.toFinset ∪ q.degrees.toFinset
⊢ x ∈ p.degrees.toFinset ∪ q.degrees.toFinset

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : DecidableEq σ
x : σ
h : Disjoint p.degrees q.degrees
hx : x ∈ p.degrees.toFinset ∪ q.degrees.toFinset
⊢ x ∈ p.degrees.toFinset ∪ q.degrees.toFinset
Module: Mathlib.Algebra.MvPolynomial.Eval

Goal state before:
pp: case refine_2.mul_X
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
p : MvPolynomial σ S₁
n : σ
ih : ↑p.coeffs ⊆ range ⇑f → p ∈ range ⇑(map f)
hx : ↑p.coeffs ⊆ range ⇑f
⊢ p * X n ∈ range ⇑(map f)

Tactic:
]

Goal state after:
case refine_2.mul_X
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
p : MvPolynomial σ S₁
n : σ
ih : ↑p.coeffs ⊆ range ⇑f → p ∈ range ⇑(map f)
hx : ↑p.coeffs ⊆ range ⇑f
⊢ p * X n ∈ range ⇑(map f)
Module: Mathlib.Algebra.MvPolynomial.Variables

Goal state before:
pp: case succ
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
φ : MvPolynomial σ R
n : ℕ
ih : (φ ^ n).vars ⊆ φ.vars
⊢ (φ * φ ^ n).vars ⊆ φ.vars

Tactic:
]

Goal state after:
case succ
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
φ : MvPolynomial σ R
n : ℕ
ih : (φ ^ n).vars ⊆ φ.vars
⊢ (φ * φ ^ n).vars ⊆ φ.vars
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
i : σ
f : MvPolynomial σ R
m : σ →₀ ℕ
h_m : m ∈ f.support
⊢ m i ≤ f.support.sup fun m ↦ m i

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
i : σ
f : MvPolynomial σ R
m : σ →₀ ℕ
h_m : m ∈ f.support
⊢ m i ≤ f.support.sup fun m ↦ m i
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝ : CommSemiring R
s : σ →₀ ℕ
i : σ
a : R
ha : a ≠ 0
⊢ s i = s i

Tactic:
]

Goal state after:
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 ∈ if c = 0 then ∅ else {d}
⊢ x ∈ p.support.erase d ∪ {d'}

Tactic:
]

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.Eval

Goal state before:
pp: case refine_2.monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
a : σ →₀ ℕ
s : S₁
p : MvPolynomial σ S₁
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ↑p.coeffs ⊆ range ⇑f → p ∈ range ⇑(map f)
ih : ↑((monomial a) s).coeffs ⊆ range ⇑f → (monomial a) s ∈ range ⇑(map f)
hx : ↑(((monomial a) s).coeffs ∪ p.coeffs) ⊆ range ⇑f
⊢ (monomial a) s + p ∈ range ⇑(map f)

Tactic:
]

Goal state after:
case refine_2.monomial_add
R : Type u
S₁ : Type v
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
a : σ →₀ ℕ
s : S₁
p : MvPolynomial σ S₁
ha : a ∉ p.coeff.support
hs : s ≠ 0
hp : ↑p.coeffs ⊆ range ⇑f → p ∈ range ⇑(map f)
ih : ↑((monomial a) s).coeffs ⊆ range ⇑f → (monomial a) s ∈ range ⇑(map f)
hx : ↑(((monomial a) s).coeffs ∪ p.coeffs) ⊆ range ⇑f
⊢ (monomial a) s + p ∈ range ⇑(map f)
Module: Mathlib.Algebra.MvPolynomial.Degrees

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i : σ
k : ℕ
⊢ k = k

Tactic:
]

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

Goal state before:
pp: case inr
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
i : σ
s : σ →₀ ℕ
h : s ∈ p.support
si : 0 < s i
⊢ ∃ b ∈ p.support, s i ≤ b i

Tactic:
]

Goal state after:
case inr
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p : MvPolynomial σ R
i : σ
s : σ →₀ ℕ
h : s ∈ p.support
si : 0 < s i
⊢ ∃ b ∈ p.support, s i ≤ b i