Theorem: FreeCMonoid.degree_mul
Proof state before:
α : Type u
inst✝³ : LinearLT α
inst✝² : DecidableRel LT.lt
deg : α → Nat
inst✝¹ : LinearLT α
inst✝ : DecidableRel LT.lt
xvars yvars✝ : KVChain LT.lt fun x => valType
x₁ : α
x₂ : valType
xs : KVChain LT.lt fun x => valType
hxs : KVChain.boundLeft x₁ xs
h_ind_xs :
  ∀ (x : KVChain LT.lt fun x => valType),
    degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) xs x } =
      degree deg { vars := xs } + degree deg { vars := x }
yvars : KVChain LT.lt fun x => valType
y₁ : α
y₂ : valType
ys : KVChain LT.lt fun x => valType
hys : KVChain.boundLeft y₁ ys
h_ind_ys :
  degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) ys } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := ys }
⊢ degree deg
      {
        vars :=
          KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) (KVChain.cons y₁ y₂ ys hys) } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := KVChain.cons y₁ y₂ ys hys }
Tactic:
apply dite (x₁ < y₁) <;> intro hx₁y₁
Proof state after:
case t
α : Type u
inst✝³ : LinearLT α
inst✝² : DecidableRel LT.lt
deg : α → Nat
inst✝¹ : LinearLT α
inst✝ : DecidableRel LT.lt
xvars yvars✝ : KVChain LT.lt fun x => valType
x₁ : α
x₂ : valType
xs : KVChain LT.lt fun x => valType
hxs : KVChain.boundLeft x₁ xs
h_ind_xs :
  ∀ (x : KVChain LT.lt fun x => valType),
    degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) xs x } =
      degree deg { vars := xs } + degree deg { vars := x }
yvars : KVChain LT.lt fun x => valType
y₁ : α
y₂ : valType
ys : KVChain LT.lt fun x => valType
hys : KVChain.boundLeft y₁ ys
h_ind_ys :
  degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) ys } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := ys }
hx₁y₁ : x₁ < y₁
⊢ degree deg
      {
        vars :=
          KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) (KVChain.cons y₁ y₂ ys hys) } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := KVChain.cons y₁ y₂ ys hys }

case e
α : Type u
inst✝³ : LinearLT α
inst✝² : DecidableRel LT.lt
deg : α → Nat
inst✝¹ : LinearLT α
inst✝ : DecidableRel LT.lt
xvars yvars✝ : KVChain LT.lt fun x => valType
x₁ : α
x₂ : valType
xs : KVChain LT.lt fun x => valType
hxs : KVChain.boundLeft x₁ xs
h_ind_xs :
  ∀ (x : KVChain LT.lt fun x => valType),
    degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) xs x } =
      degree deg { vars := xs } + degree deg { vars := x }
yvars : KVChain LT.lt fun x => valType
y₁ : α
y₂ : valType
ys : KVChain LT.lt fun x => valType
hys : KVChain.boundLeft y₁ ys
h_ind_ys :
  degree deg { vars := KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) ys } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := ys }
hx₁y₁ : ¬x₁ < y₁
⊢ degree deg
      {
        vars :=
          KVChain.mergeWith (fun x a b => some (valAdd a b)) (KVChain.cons x₁ x₂ xs hxs) (KVChain.cons y₁ y₂ ys hys) } =
    degree deg { vars := KVChain.cons x₁ x₂ xs hxs } + degree deg { vars := KVChain.cons y₁ y₂ ys hys }