-- Mathlib/RingTheory/Polynomial/SeparableDegree.lean
theorem IsSeparableContraction.degree_eq [hF : ExpChar F q] (g : F[X])
    (hg : IsSeparableContraction q f g) : g.natDegree = hf.degree

Proof state:
case zero.intro.intro
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
left✝ : g.Separable
m : ℕ
hm : g = f
⊢ g.natDegree = hf.degree

Tactic:
rw [hf.eq_degree, hm]

Result:
no goals

Proof state:
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
hF : ExpChar F q
g : F[X]
hg : IsSeparableContraction q f g
⊢ g.natDegree = hf.degree

Tactic:
cases hF

Result:
case zero
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
hg : IsSeparableContraction 1 f g
⊢ g.natDegree = hf.degree

case prime
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hg : IsSeparableContraction q f g
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
⊢ g.natDegree = hf.degree

Proof state:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
⊢ g.natDegree = hf.degree

Tactic:
haveI : Fact q.Prime := ⟨by assumption⟩

Result:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
this : Fact (Nat.Prime q)
⊢ g.natDegree = hf.degree

Proof state:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
this : Fact (Nat.Prime q)
⊢ g.natDegree = hf.degree

Tactic:
refine contraction_degree_eq_or_insep q g g' m m' ?_ hg hg'

Result:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
this : Fact (Nat.Prime q)
⊢ (expand F (q ^ m)) g = (expand F (q ^ m')) g'

Proof state:
case zero
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
hg : IsSeparableContraction 1 f g
⊢ g.natDegree = hf.degree

Tactic:
rcases hg with ⟨_, m, hm⟩

Result:
case zero.intro.intro
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
left✝ : g.Separable
m : ℕ
hm : (expand F (1 ^ m)) g = f
⊢ g.natDegree = hf.degree

Proof state:
case zero.intro.intro
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
left✝ : g.Separable
m : ℕ
hm : (expand F (1 ^ m)) g = f
⊢ g.natDegree = hf.degree

Tactic:
rw [one_pow, expand_one] at hm

Result:
case zero.intro.intro
F : Type u_1
inst✝¹ : Field F
f g : F[X]
inst✝ : CharZero F
hf : HasSeparableContraction 1 f
left✝ : g.Separable
m : ℕ
hm : g = f
⊢ g.natDegree = hf.degree

Proof state:
case prime.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
⊢ g.natDegree = hf.degree

Tactic:
obtain ⟨hg', m', hm'⟩ := Classical.choose_spec hf

Result:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
⊢ g.natDegree = hf.degree

Proof state:
case prime.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
⊢ g.natDegree = hf.degree

Tactic:
let g' := Classical.choose hf

Result:
case prime.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
⊢ g.natDegree = hf.degree

Proof state:
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
⊢ Nat.Prime q

Tactic:
assumption

Result:
no goals

Proof state:
case prime
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hg : IsSeparableContraction q f g
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
⊢ g.natDegree = hf.degree

Tactic:
rcases hg with ⟨hg, m, hm⟩

Result:
case prime.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
⊢ g.natDegree = hf.degree

Proof state:
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
this : Fact (Nat.Prime q)
⊢ (expand F (q ^ m)) g = (expand F (q ^ m')) g'

Tactic:
rw [hm, hm']

Result:
no goals