internlm/Lean-Github huggingface dataset:
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.left.intro.intro.intro.intro.inl
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_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 ())
R₁ R₂ : β
oldWorld₁ : (collection R₁).fst
oldWorld₂ : (collection R₂).fst
hneq : R₁ ≠ R₂
IH_rel :
  ∀ (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
sameR :
  ∀ (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⟩)
noNewL :
  ∀ (R : β) (oldWorld : (collection R).fst),
    ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
u_new : Unit
a_rel_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inl u_new)
right✝ : relate (combinedModel collection newVal).1 b (Sum.inl u_new) (Sum.inr ⟨R₂, oldWorld₂⟩)
⊢ ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inl u_new)
Tactic:
apply noNewL
Proof state after:
no goals
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.left.intro.intro.intro.intro.inr.mk
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_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 ())
R₁ R₂ : β
oldWorld₁ : (collection R₁).fst
oldWorld₂ : (collection R₂).fst
hneq : R₁ ≠ R₂
IH_rel :
  ∀ (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
noNewL :
  ∀ (R : β) (oldWorld : (collection R).fst),
    ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
otherR : β
otherWorld : (collection otherR).fst
a_rel_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨otherR, otherWorld⟩)
right✝ : relate (combinedModel collection newVal).1 b (Sum.inr ⟨otherR, otherWorld⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)
sameR :
  R₁ ≠ otherR → ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨otherR, otherWorld⟩)
⊢ False
Tactic:
simp_all
Proof state after:
no goals
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.right
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),
    ¬relate (combinedModel collection newVal).1 (a;'b) (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
Tactic:
intro R oldWorld
Proof state after:
case right.right
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
⊢ ¬relate (combinedModel collection newVal).1 (a;'b) (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.right
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
⊢ ¬relate (combinedModel collection newVal).1 (a;'b) (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
Tactic:
unfold relate
Proof state after:
case right.right
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
⊢ ¬∃ y,
      relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) y ∧
        relate (combinedModel collection newVal).1 b y (Sum.inl ())
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.right
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
⊢ ¬∃ y,
      relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) y ∧
        relate (combinedModel collection newVal).1 b y (Sum.inl ())
Tactic:
rintro ⟨u, a_rel_to_u, b_rel_u_to⟩
Proof state after:
case right.right.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
oldWorld : (collection R).fst
u : Unit ⊕ (k : β) × (collection k).fst
a_rel_to_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) u
b_rel_u_to : relate (combinedModel collection newVal).1 b u (Sum.inl ())
⊢ False
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
case right.right.intro.intro.inl.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_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✝ : Unit
a_rel_to_u : relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl val✝)
b_rel_u_to : relate (combinedModel collection newVal).1 b (Sum.inl val✝) (Sum.inl ())
IH_rel :
  ∀ (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
sameR :
  ∀ (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⟩)
noNewL :
  ∀ (R : β) (oldWorld : (collection R).fst),
    ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())
⊢ ¬relate (combinedModel collection newVal).1 a (Sum.inr ⟨R, oldWorld⟩) (Sum.inl val✝)
Tactic:
apply noNewL
Proof state after:
no goals
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
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 ())
Tactic:
apply noNewL
Proof state after:
no goals
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
⊢ ∀ (a a_1 : Program),
    (fun π =>
          (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
              relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
                relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
            (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
                R₁ ≠ R₂ →
                  ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
              ∀ (R : β) (oldWorld : (collection R).fst),
                ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
        a →
      (fun π =>
            (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
                relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
                  relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
              (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
                  R₁ ≠ R₂ →
                    ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
                ∀ (R : β) (oldWorld : (collection R).fst),
                  ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
          a_1 →
        (fun π =>
            (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
                relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
                  relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
              (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
                  R₁ ≠ R₂ →
                    ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
                ∀ (R : β) (oldWorld : (collection R).fst),
                  ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
          (a⋓a_1)
Tactic:
aesop
Proof state after:
no goals
Theorem: combMo_preserves_truth_at_oldWOrld
Proof state before:
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
⊢ ∀ (a : Program),
    (fun π =>
          (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
              relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
                relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
            (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
                R₁ ≠ R₂ →
                  ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
              ∀ (R : β) (oldWorld : (collection R).fst),
                ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
        a →
      (fun π =>
          (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
              relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
                relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
            (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
                R₁ ≠ R₂ →
                  ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
              ∀ (R : β) (oldWorld : (collection R).fst),
                ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
        (∗a)
Tactic:
intro a IH_a
Proof state after:
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
a : 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) (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 ())