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8 Some PITL theorems and Their Proofs

This appendix gives a representative set of PITL theorems and derived inference rules together with their proofs. Many are used either directly or indirectly in the completeness proof for PITL with both finite and infinite time. We have partially organised the material, particularly in Section 8.2, along the lines of some standard modal logic systems.

The PITL theorems and derived rules have a shared index sequence (e.g., BfChopImpChopBoxChopEqvChop are followed by BfGen rather than DR1). We believe that this convention simplifies locating material in this appendix.

Proof steps can refer to axioms, inference rules, previously deduced theorems, derived inference rules and also the following:

  • Assumptions which are regarded as being previously deduced.
  • Prop: Conventional nonmodal propositional reasoning (by restricted application of Axiom VPTL) and Modus Ponens.
  • ImpChain: A chain of implications.
  • EqvChain: A chain of equivalences.
  • In principle, ImpChain and EqvChain are subsumed by Prop but are used here to make the reasoning more explicit.

Our assumption of axiomatic completeness for PITL with just finite time PITLF
permits any valid implication of the form finite f.

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