Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case left.mp
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
oldWorld₁ oldWorld₂ : (collection R).fst
new_rel :
  ∃ y,
    relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld₁⟩) y ∧
      relate (combinedModel collection newVal).1 b y (Sum.inr ⟨R, oldWorld₂⟩)
⊢ relate (collection R).snd.2.1 (a;'b) oldWorld₁ oldWorld₂
Tactic:
rcases new_rel with ⟨u, a_rel_u, u_rel_b⟩
Proof state after:
case left.mp.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 ())
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
oldWorld₁ oldWorld₂ : (collection R).fst
u : Unit ⊕ (k : β) × (collection k).fst
a_rel_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld₁⟩) u
u_rel_b : relate (combinedModel collection newVal).1 b u (Sum.inr ⟨R, oldWorld₂⟩)
⊢ relate (collection R).snd.2.1 (a;'b) oldWorld₁ oldWorld₂