mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.EraseLead

Goal state before:
pp: case neg.succ
R : Type u_1
inst✝³ : Semiring R
S : Type u_2
F : Type u_3
inst✝² : Semiring S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : AddMonoidHomClass F R[X] S[X]
φ : F
p : R[X]
fu : ℕ → ℕ
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
f g : R[X]
fg : f.natDegree < g.natDegree
a✝ : g.natDegree ≤ p.natDegree
fk : (φ f).natDegree = fu f.natDegree
gk : (φ g).natDegree = fu g.natDegree
n✝ : ℕ
fu0 : ∀ {n : ℕ}, n ≤ n✝ + 1 → fu n = 0
fc : ∀ {n m : ℕ}, n✝ + 1 ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < n✝ + 1 → φ f = 0
FG : f.natDegree < n✝ + 1
⊢ (φ g).natDegree = fu g.natDegree

Tactic:
]

Goal state after:
case neg.succ
R : Type u_1
inst✝³ : Semiring R
S : Type u_2
F : Type u_3
inst✝² : Semiring S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : AddMonoidHomClass F R[X] S[X]
φ : F
p : R[X]
fu : ℕ → ℕ
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
f g : R[X]
fg : f.natDegree < g.natDegree
a✝ : g.natDegree ≤ p.natDegree
fk : (φ f).natDegree = fu f.natDegree
gk : (φ g).natDegree = fu g.natDegree
n✝ : ℕ
fu0 : ∀ {n : ℕ}, n ≤ n✝ + 1 → fu n = 0
fc : ∀ {n m : ℕ}, n✝ + 1 ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < n✝ + 1 → φ f = 0
FG : f.natDegree < n✝ + 1
⊢ (φ g).natDegree = fu g.natDegree
Module: Mathlib.Algebra.MvPolynomial.Equiv

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α β : Type} (f : α → β), (fun x ↦ x) ∘ f = f
h₁ : ∀ {α β : Type} (f : α → β), (f ∘ fun x ↦ x) = f
h' : (((finSuccEquiv R n) f).support.sup fun x ↦ x) = degreeOf 0 f
⊢ ((finSuccEquiv R n) f).support.sup WithBot.some = ((finSuccEquiv R n) f).support.sup WithBot.some

Tactic:
]

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

Goal state before:
pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff = a.leadingCoeff * (Multiset.map leadingCoeff t).prod

pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff ≠ 0

Tactic:
]

Goal state after:
case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff = a.leadingCoeff * (Multiset.map leadingCoeff t).prod

case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff ≠ 0
Module: Mathlib.Algebra.Polynomial.BigOperators

Goal state before:
pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * (Multiset.map leadingCoeff t).prod = a.leadingCoeff * (Multiset.map leadingCoeff t).prod

pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0

Tactic:
]

Goal state after:
case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0
Module: Mathlib.Algebra.MvPolynomial.Equiv

Goal state before:
pp: case pos
R : Type u
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : f = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case refine_1
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1
f g : R[X]
hfg : p = f * g
⊢ p.Monic

Tactic:
]

Goal state after:
case refine_1
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1
f g : R[X]
hfg : p = f * g
⊢ p.Monic
Module: Mathlib.Algebra.MvPolynomial.Equiv

Goal state before:
pp: case neg
R : Type u
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ degreeOf 0 f = degreeOf 0 f

Tactic:
]

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

Goal state before:
pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0

pp: case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0

Tactic:
]

Goal state after:
case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0

case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0
Module: Mathlib.Algebra.MvPolynomial.Equiv

Goal state before:
pp: R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
a : σ
p : MvPolynomial σ R
⊢ degreeOf none ((rename ⇑(Equiv.optionSubtypeNe a).symm) p) = degreeOf a p

Tactic:
]

Goal state after:
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
a : σ
p : MvPolynomial σ R
⊢ degreeOf none ((rename ⇑(Equiv.optionSubtypeNe a).symm) p) = degreeOf a p
Module: Mathlib.Algebra.Polynomial.EraseLead

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
this : (∑ i, C (x i) * X ^ k i).support = image k univ
⊢ n = n

Tactic:
]

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

Goal state before:
pp: case e'_2.e'_4
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
a : σ
p : MvPolynomial σ R
⊢ none = none

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ ∀ b ∈ (((finSuccEquiv R n) p).coeff i).support, b j ≤ p.support.sup fun m ↦ m j.succ

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ ∀ b ∈ (((finSuccEquiv R n) p).coeff i).support, b j ≤ p.support.sup fun m ↦ m j.succ
Module: Mathlib.Algebra.Polynomial.Monic

Goal state before:
pp: case refine_2
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1
f g : R[X]
hfg : p = f * g
⊢ p.Monic

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1
f g : R[X]
hfg : p = f * g
⊢ p.Monic
Module: Mathlib.Algebra.MvPolynomial.Equiv

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ (((finSuccEquiv R n) p).coeff i).support
⊢ (cons i m) j.succ ≤ p.support.sup fun m ↦ m j.succ

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ (((finSuccEquiv R n) p).coeff i).support
⊢ (cons i m) j.succ ≤ p.support.sup fun m ↦ m j.succ
Module: Mathlib.Algebra.Polynomial.EraseLead

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
⊢ (∑ i, C (x i) * X ^ k i).support = image k univ

Tactic:
simp_rw [Finset.ext_iff, mem_support_iff, finsetSum_coeff, coeff_C_mul_X_pow, mem_image,
    mem_univ, true_and]

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
⊢ ∀ (a : ℕ), (∑ x_1, if a = k x_1 then x x_1 else 0) ≠ 0 ↔ ∃ a_1, k a_1 = a
Module: Mathlib.Algebra.Polynomial.BigOperators

Goal state before:
pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (a * t.prod).natDegree = (a.natDegree ::ₘ Multiset.map (fun f ↦ f.natDegree) t).sum

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (a * t.prod).natDegree = (a.natDegree ::ₘ Multiset.map (fun f ↦ f.natDegree) t).sum
Module: Mathlib.Algebra.Polynomial.Monic

Goal state before:
pp: case refine_3
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1
f g : R[X]
hfg : p = f * g
⊢ p = p

Tactic:
]

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

Goal state before:
pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ a.natDegree + (Multiset.map (fun f ↦ f.natDegree) t).sum = a.natDegree + (Multiset.map (fun f ↦ f.natDegree) t).sum

pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0

pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff ≠ 0

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0

case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff ≠ 0
Module: Mathlib.Algebra.Polynomial.Monic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : ¬p = 1
⊢ (∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1) ↔
    ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f.natDegree = 0 ∨ g.natDegree = 0

Tactic:
]

Goal state after:
case neg
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : ¬p = 1
⊢ (∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1) ↔
    ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f.natDegree = 0 ∨ g.natDegree = 0
Module: Mathlib.Algebra.Polynomial.EraseLead

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
j : Fin n
⊢ x j ≠ 0

pp: case refine_2
R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
j : Fin n
⊢ ∀ b ∈ univ, b ≠ j → (if k j = k b then x b else 0) = 0

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
j : Fin n
⊢ x j ≠ 0

case refine_2
R : Type u_1
inst✝ : Semiring R
n : ℕ
k : Fin n → ℕ
x : Fin n → R
hk : Function.Injective k
hx : ∀ (i : Fin n), x i ≠ 0
j : Fin n
⊢ ∀ b ∈ univ, b ≠ j → (if k j = k b then x b else 0) = 0
Module: Mathlib.Algebra.Polynomial.BigOperators

Goal state before:
pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0

pp: case refine_2.h
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0

case refine_2.h
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum
ht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0
Module: Mathlib.Algebra.Polynomial.Monic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : ¬p = 1
a b : R[X]
ha : a.Monic
hb : b.Monic
⊢ a * b = p → a = 1 ∨ b = 1 ↔ a * b = p → a = 1 ∨ b = 1

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
σ : Type u_2
τ : Type u_3
f : σ → τ
i : σ
p : R[X]
⊢ (Polynomial.aeval (X (f i))) p = (Polynomial.aeval (X (f i))) p

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
t : Multiset R[X]
inst✝ : Nontrivial R
h : ∀ f ∈ t, f.Monic
this : t.prod ≠ 0
⊢ (Multiset.map degree t).sum = (Multiset.map degree t).sum

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
t : Multiset R[X]
inst✝ : Nontrivial R
h : ∀ f ∈ t, f.Monic
this : t.prod ≠ 0
⊢ t.prod.degree = (Multiset.map degree t).sum

Tactic:
rw [degree_eq_natDegree this, natDegree_multiset_prod_of_monic _ h, Nat.cast_multiset_sum,
    Multiset.map_map, Function.comp_def,
    Multiset.map_congr rfl (fun f hf => (degree_eq_natDegree (h f hf).ne_zero).symm)]

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

Goal state before:
pp: R : Type u_1
σ : Type u_3
inst✝ : CommSemiring R
i : σ
p q : R[X]
h : p = q
⊢ (toMvPolynomial i) q = (toMvPolynomial i) q

Tactic:
]

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

Goal state before:
pp: case succ.refine_1
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
j : Fin (n + 1)
i : Fin n
hij : i.castSucc < j
⊢ k i < Function.extend Fin.castSucc k (fun x ↦ f.natDegree) j

Tactic:
]

Goal state after:
case succ.refine_1
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
j : Fin (n + 1)
i : Fin n
hij : i.castSucc < j
⊢ k i < Function.extend Fin.castSucc k (fun x ↦ f.natDegree) j
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ 0 ^ (n + 1) + ∑ i ∈ range (n + 1), 0 ^ i = if n + 2 = 0 then 0 else 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ 0 ^ (n + 1) + ∑ i ∈ range (n + 1), 0 ^ i = if n + 2 = 0 then 0 else 1
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ ∑ i ∈ range (n + 2), 0 ^ i = if n + 2 = 0 then 0 else 1

Tactic:
rw [geom_sum_succ']

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ 0 ^ (n + 1) + ∑ i ∈ range (n + 1), 0 ^ i = if n + 2 = 0 then 0 else 1
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
x y : R
n : ℕ
⊢ ∑ j ∈ range n, op y ^ (n - 1 - (n - 1 - j)) * op x ^ (n - 1 - j) = ∑ i ∈ range n, op y ^ i * op x ^ (n - 1 - i)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
x y : R
n : ℕ
⊢ ∑ j ∈ range n, op y ^ (n - 1 - (n - 1 - j)) * op x ^ (n - 1 - j) = ∑ i ∈ range n, op y ^ i * op x ^ (n - 1 - i)
Module: Mathlib.Algebra.Polynomial.Monic

Goal state before:
pp: case mp
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f g : R[X]
hf : f.Monic
hg : g.Monic
hp : (f * g).Monic
h : ∀ (f_1 g_1 : R[X]), f_1.Monic → g_1.Monic → f_1 * g_1 = f * g → f_1.natDegree = 0 ∨ g_1.natDegree = 0
⊢ ¬(0 < g.natDegree ∧ g.natDegree ≤ f.natDegree)

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f g : R[X]
hf : f.Monic
hg : g.Monic
hp : (f * g).Monic
h : ∀ (f_1 g_1 : R[X]), f_1.Monic → g_1.Monic → f_1 * g_1 = f * g → f_1.natDegree = 0 ∨ g_1.natDegree = 0
⊢ ¬(0 < g.natDegree ∧ g.natDegree ≤ f.natDegree)
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
x : R
n i : ℕ
x✝ : i ∈ range n
⊢ x ^ i = x ^ i

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
i j : Fin n
hij : i.castSucc < j.castSucc
⊢ i.castSucc < j.castSucc

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
i j : Fin n
hij : i.castSucc < j.castSucc
⊢ i.castSucc < j.castSucc
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: case zero
R : Type u_1
inst✝ : Semiring R
x y : R
h : Commute x y
f : ℕ → ℕ → R := fun m i ↦ (x + y) ^ i * y ^ (m - 1 - i)
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
x y : R
h : Commute x y
f : ℕ → ℕ → R := fun m i ↦ (x + y) ^ i * y ^ (m - 1 - i)
n : ℕ
ih : (∑ i ∈ range n, f n i) * x + y ^ n = (x + y) ^ n
⊢ (x + y) ^ n = (x + y) ^ n

Tactic:
]

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

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
j : Fin (n + 1)
i : Fin n
hij : i.castSucc < j
hj : ¬∃ j₀, j₀.castSucc = j
⊢ k i < f.natDegree

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
n : ℕ
hn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i
f : R[X]
h : #f.support = n + 1
k : Fin n → ℕ
x : Fin n → R
hk : StrictMono k
hx : ∀ (i : Fin n), x i ≠ 0
hf : f.eraseLead = ∑ i, C (x i) * X ^ k i
H : ¬∃ k, k.castSucc = Fin.last n
j : Fin (n + 1)
i : Fin n
hij : i.castSucc < j
hj : ¬∃ j₀, j₀.castSucc = j
⊢ k i < f.natDegree
Module: Mathlib.Algebra.Polynomial.BigOperators

Goal state before:
pp: case refine_2
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ht : t.prod.coeff 0 = (Multiset.map (fun f ↦ f.coeff 0) t).prod
⊢ a.coeff 0 * (Multiset.map (fun f ↦ f.coeff 0) t).prod = a.coeff 0 * (Multiset.map (fun f ↦ f.coeff 0) t).prod

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
⊢ (∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2)) ↔
    ∀ (q : R[X]), q.Monic → q.natDegree ∈ Ioc 0 (p.natDegree / 2) → ¬q ∣ p

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p : R[X]
hp : p.Monic
hp1 : p ≠ 1
⊢ (∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2)) ↔
    ∀ (q : R[X]), q.Monic → q.natDegree ∈ Ioc 0 (p.natDegree / 2) → ¬q ∣ p
Module: Mathlib.Algebra.Polynomial.BigOperators

Goal state before:
pp: R : Type u
inst✝ : CommRing R
t : Multiset R
⊢ (Multiset.map (fun i ↦ (X - C i).nextCoeff) t).sum = -t.sum

pp: case h
R : Type u
inst✝ : CommRing R
t : Multiset R
⊢ ∀ i ∈ t, (X - C i).Monic

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
t : Multiset R
⊢ (Multiset.map (fun i ↦ (X - C i).nextCoeff) t).sum = -t.sum

case h
R : Type u
inst✝ : CommRing R
t : Multiset R
⊢ ∀ i ∈ t, (X - C i).Monic
Module: Mathlib.Algebra.Ring.GeomSum

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
x y : R
h : Commute x y
f : ℕ → ℕ → R := fun m i ↦ (x + y) ^ i * y ^ (m - 1 - i)
n : ℕ
ih : (∑ i ∈ range n, f n i) * x + y ^ n = (x + y) ^ n
f_last : f (n + 1) n = (x + y) ^ n
i : ℕ
hi : i ∈ range n
this : Commute y ((x + y) ^ i)
⊢ (x + y) ^ i * y ^ (n - i) = (x + y) ^ i * y ^ (n - (i + 1) + 1)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
x y : R
h : Commute x y
f : ℕ → ℕ → R := fun m i ↦ (x + y) ^ i * y ^ (m - 1 - i)
n : ℕ
ih : (∑ i ∈ range n, f n i) * x + y ^ n = (x + y) ^ n
f_last : f (n + 1) n = (x + y) ^ n
i : ℕ
hi : i ∈ range n
this : Commute y ((x + y) ^ i)
⊢ (x + y) ^ i * y ^ (n - i) = (x + y) ^ i * y ^ (n - (i + 1) + 1)