| ⊢ | Substitution instances of valid PTL formulas |  | VPTL | 
| ⊢ | (f ⌢ g) ⌢ h ≡ f ⌢ (g ⌢ h) |  | ChopAssoc | 
| ⊢ | (f0 ∨ f1) ⌢ g  ⊃ (f0 ⌢ g) ∨ (f1 ⌢ g) |  | OrChopImp | 
| ⊢ | f ⌢ (g0 ∨ g1)  ⊃ (f ⌢ g0) ∨ (f ⌢ g1) |  | ChopOrImp | 
| ⊢ | empty ⌢ f ≡ f |  | EmptyChop | 
| ⊢ | finite ⊃ (f ⌢ empty ≡ f) |  | FiniteImpChopEmpty | 
| ⊢ | w  ⊃  w |  | StateImpBf | 
| ⊢ | (f0  ⊃ f1) ∧ (g0  ⊃ g1)  ⊃ (f0 ⌢ g0)  ⊃ (f1 ⌢ g1) |  | BfAndBoxImpChopImpChop | 
| ⊢ | f⋆ ≡empty ∨ (f ∧more) ⌢ f⋆ |  | SChopStarEqv | 
| ⊢ | f ∧ (f  ⊃ (g ∧more) ⌢ f)  ⊃ gω |  | ChopOmegaInduct | 
| ⊢ | f  ⊃ g,   ⊢ f   ⇒ ⊢ g |  | MP | 
| ⊢ | finite ⊃ f   ⇒ ⊢ f |  | BfFGen | 
| ⊢ | f   ⇒ ⊢ f |  | BoxGen | 
| ⊢ | ((fin P) ≡ g) ⊃ f   ⇒ ⊢ f |  | BfAux |