Theorem: Polynomial.mono_map_natDegree_eq
File: Mathlib/Algebra/Polynomial/EraseLead.lean

tactic: refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)
  p.natDegree (by simp [fu0]) ?_ ?_ _ rfl.le
state_before: R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ (φ p).natDegree = fu p.natDegree
state_after: case refine_1
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ p.natDegree → (fun p => (φ p).natDegree = fu p.natDegree) (C r * X ^ n)

case refine_2
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ ∀ (f g : R[X]),
    f.natDegree < g.natDegree →
      g.natDegree ≤ p.natDegree →
        (fun p => (φ p).natDegree = fu p.natDegree) f →
          (fun p => (φ p).natDegree = fu p.natDegree) g → (fun p => (φ p).natDegree = fu p.natDegree) (f + g)

tactic: simp [fu0]
state_before: R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ (fun p => (φ p).natDegree = fu p.natDegree) 0
state_after: no goals

tactic: intro n r r0 _
state_before: case refine_1
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ p.natDegree → (fun p => (φ p).natDegree = fu p.natDegree) (C r * X ^ n)
state_after: case refine_1
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
n : ℕ
r : R
r0 : r ≠ 0
a✝ : n ≤ p.natDegree
⊢ (φ (C r * X ^ n)).natDegree = fu (C r * X ^ n).natDegree

tactic: rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]
state_before: case refine_1
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
n : ℕ
r : R
r0 : r ≠ 0
a✝ : n ≤ p.natDegree
⊢ (φ (C r * X ^ n)).natDegree = fu (C r * X ^ n).natDegree
state_after: no goals

tactic: intro f g fg _ fk gk
state_before: case refine_2
R : Type u_1
inst✝³ : Semiring R
f : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n
⊢ ∀ (f g : R[X]),
    f.natDegree < g.natDegree →
      g.natDegree ≤ p.natDegree →
        (fun p => (φ p).natDegree = fu p.natDegree) f →
          (fun p => (φ p).natDegree = fu p.natDegree) g → (fun p => (φ p).natDegree = fu p.natDegree) (f + g)
state_after: case refine_2
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
⊢ (φ (f + g)).natDegree = fu (f + g).natDegree

tactic: rw [natDegree_add_eq_right_of_natDegree_lt fg, _root_.map_add]
state_before: case refine_2
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
⊢ (φ (f + g)).natDegree = fu (f + g).natDegree
state_after: case refine_2
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
⊢ (φ f + φ g).natDegree = fu g.natDegree

tactic: by_cases FG : k ≤ f.natDegree
state_before: case refine_2
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
⊢ (φ f + φ g).natDegree = fu g.natDegree
state_after: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree

case neg
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : ¬k ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree

tactic: rw [natDegree_add_eq_right_of_natDegree_lt, gk]
state_before: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree
state_after: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ (φ f).natDegree < (φ g).natDegree

tactic: rw [fk, gk]
state_before: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ (φ f).natDegree < (φ g).natDegree
state_after: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ fu f.natDegree < fu g.natDegree

tactic: exact fc FG fg
state_before: case pos
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : k ≤ f.natDegree
⊢ fu f.natDegree < fu g.natDegree
state_after: no goals

tactic: cases k
state_before: case neg
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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]
k : ℕ
fu : ℕ → ℕ
fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0
fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0
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
FG : ¬k ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree
state_after: case neg.zero
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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
fu0 : ∀ {n : ℕ}, n ≤ 0 → fu n = 0
fc : ∀ {n m : ℕ}, 0 ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < 0 → φ f = 0
FG : ¬0 ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree

case neg.succ
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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 : ¬n✝ + 1 ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree

tactic: exact (FG (Nat.zero_le _)).elim
state_before: case neg.zero
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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
fu0 : ∀ {n : ℕ}, n ≤ 0 → fu n = 0
fc : ∀ {n m : ℕ}, 0 ≤ n → n < m → fu n < fu m
φ_k : ∀ {f : R[X]}, f.natDegree < 0 → φ f = 0
FG : ¬0 ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree
state_after: no goals

tactic: rwa [φ_k (not_le.mp FG), zero_add]
state_before: case neg.succ
R : Type u_1
inst✝³ : Semiring R
f✝ : R[X]
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 : ¬n✝ + 1 ≤ f.natDegree
⊢ (φ f + φ g).natDegree = fu g.natDegree
state_after: no goals