rcases IH_b with ⟨IH_rel, sameR, noNewL⟩
case right.right.intro.intro.inr
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
a b : Program
IH_a :
  (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
      relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
        relate (collection R).snd.2.1 a oldWorld₁ oldWorld₂) ∧
    (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
        R₁ ≠ R₂ → ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
      ∀ (R : β) (oldWorld : (collection R).fst),
        ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
IH_b :
  (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
      relate (combinedModel collection newVal).1 b (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
        relate (collection R).snd.2.1 b oldWorld₁ oldWorld₂) ∧
    (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
        R₁ ≠ R₂ → ¬relate (combinedModel collection newVal).1 b (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
      ∀ (R : β) (oldWorld : (collection R).fst),
        ¬relate (combinedModel collection newVal).1 b (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
R : β
oldWorld : (collection R).fst
val✝ : (k : β) × (collection k).fst
a_rel_to_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inr val✝)
b_rel_u_to : relate (combinedModel collection newVal).1 b (Sum.inr val✝) (Sum.inl ())
⊢ ¬relate (combinedModel collection newVal).1 b (Sum.inr val✝) (Sum.inl ())
case right.right.intro.intro.inr.intro.intro
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
a b : Program
IH_a :
  (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
      relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
        relate (collection R).snd.2.1 a oldWorld₁ oldWorld₂) ∧
    (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
        R₁ ≠ R₂ → ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
      ∀ (R : β) (oldWorld : (collection R).fst),
        ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
R : β
oldWorld : (collection R).fst
val✝ : (k : β) × (collection k).fst
a_rel_to_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inr val✝)
b_rel_u_to : relate (combinedModel collection newVal).1 b (Sum.inr val✝) (Sum.inl ())
IH_rel :
  ∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
    relate (combinedModel collection newVal).1 b (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
      relate (collection R).snd.2.1 b oldWorld₁ oldWorld₂
sameR :
  ∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
    R₁ ≠ R₂ → ¬relate (combinedModel collection newVal).1 b (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)
noNewL :
  ∀ (R : β) (oldWorld : (collection R).fst),
    ¬relate (combinedModel collection newVal).1 b (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
⊢ ¬relate (combinedModel collection newVal).1 b (Sum.inr val✝) (Sum.inl ())