logical foundations of cyber physical systems
play

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr - PowerPoint PPT Presentation

Mar 27, 2024 •309 likes •1.77k views

06: Truth & Proof Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 1 / 23 Outline Learning Objectives

  1. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ → L: assume RHS of an assumed implication after proving its LHS André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  2. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ ¬ R Γ ⊢ ¬ P , ∆ Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  3. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  4. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ ¬ R: prove ¬ P by proving contradiction (or ∆ options) from assumption P André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  5. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ ¬ L Γ , ¬ P ⊢ ∆ Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  6. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  7. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ ¬ L: assume ¬ P by proving its opposite P André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  8. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  9. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ id: proof done (marked ∗ ) when succedent to prove is in antecedent André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  10. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ Γ , P → Q ⊢ ∆ id: only way to finish a proof (in propositional logic!) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  11. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ , P → Q ⊢ ∆ Γ ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  12. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  13. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ cut: Show lemma C and then assume lemma C André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  14. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  15. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ ⊤ R: proof done (marked ∗ ) when proving trivial true (used rarely) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  16. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ ⊤ R: what rule to use when true in antecedent? André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  17. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ ⊥ L Γ , false ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  18. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ ⊥ L Γ , false ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ ⊥ L: proof done (marked ∗ ) when assuming trivial false (used rarely) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  19. Propositional Proof Rules of Sequent Calculus ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ ∨ R Γ ⊢ P , Q , ∆ Γ , P ⊢ ∆ ¬ R Γ ⊢ P ∧ Q , ∆ Γ ⊢ P ∨ Q , ∆ Γ ⊢ ¬ P , ∆ ∧ L Γ , P , Q ⊢ ∆ ∨ L Γ , P ⊢ ∆ Γ , Q ⊢ ∆ Γ ⊢ P , ∆ ¬ L Γ , P ∧ Q ⊢ ∆ Γ , P ∨ Q ⊢ ∆ Γ , ¬ P ⊢ ∆ Γ , P ⊢ Q , ∆ → R id Γ , P ⊢ P , ∆ ⊤ R Γ ⊢ true , ∆ Γ ⊢ P → Q , ∆ → L Γ ⊢ P , ∆ Γ , Q ⊢ ∆ cut Γ ⊢ C , ∆ Γ , C ⊢ ∆ ⊥ L Γ , false ⊢ ∆ Γ , P → Q ⊢ ∆ Γ ⊢ ∆ ⊥ L: what rule to use when false in succedent? André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 6 / 23

  20. Sequent Proof Example (Simple) ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  21. Sequent Proof Example (Simple) v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  22. Sequent Proof Example (Simple) ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  23. Sequent Proof Example (Simple) ∧ L v 2 ≤ 10 , b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  24. Sequent Proof Example (Simple) ∗ id v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  25. Sequent Proof Example (Simple) ∗ id ∨ R v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  26. Sequent Proof Example (Simple) ∗ ∧ L v 2 ≤ 10 , b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 id v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L ∨ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  27. Sequent Proof Example (Simple) ∗ id ∗ v 2 ≤ 10 , b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 ∧ L id v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L ∨ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 7 / 23

  28. Soundness of Proof Rules Lemma ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ is sound Γ ⊢ P ∧ Q , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 8 / 23

  29. Soundness of Proof Rules Lemma ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ is sound: conclusion valid if all premises valid. Γ ⊢ P ∧ Q , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 8 / 23

  30. Soundness of Proof Rules Lemma ∧ R Γ ⊢ P , ∆ Γ ⊢ Q , ∆ is sound: conclusion valid if all premises valid. Γ ⊢ P ∧ Q , ∆ using [ [ P ∧ Q ] ] = [ [ P ] ] ∩ [ [ Q ] ] . Proof ω ∈ [ [ G ] ] for all G ∈ Γ and ω �∈ [ [ D ] ] for all D ∈ ∆ WLOG: (why?) By premise: ω ∈ [ [Γ ⊢ P , ∆] ] and ω ∈ [ [Γ ⊢ Q , ∆] ] ω ∈ [ [ P ] ] and ω ∈ [ [ Q ] ] By WLOG: By semantics: ω ∈ [ [ P ∧ Q ] ] By definition: ω ∈ [ [Γ ⊢ P ∧ Q , ∆] ] André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 8 / 23

  31. Soundness of dL Theorem dL sequent calculus is sound: every dL formula with a proof is valid. André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 9 / 23

  32. Soundness of dL Theorem dL sequent calculus is sound: every dL sequent with a proof is valid. Proof (by induction on structure of sequent calculus proof). André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 9 / 23

  33. Soundness of dL Theorem dL sequent calculus is sound: every dL sequent with a proof is valid. Proof (by induction on structure of sequent calculus proof). Proofs without rule uses only prove dL axioms, which are sound. 0 André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 9 / 23

  34. Soundness of dL Theorem dL sequent calculus is sound: every dL sequent with a proof is valid. Proof (by induction on structure of sequent calculus proof). Proofs without rule uses only prove dL axioms, which are sound. 0 Sequent proof ends with some proof step: 1 Γ 1 ⊢ ∆ 1 Γ n ⊢ ∆ n ... Γ ⊢ ∆ The subproof of each premise Γ i ⊢ ∆ i is smaller, so � Γ i ⊢ ∆ i by IH. All dL proof rules are proved sound, also the one used above, i.e.: If � (Γ 1 ⊢ ∆ 1 ) and ... and � (Γ n ⊢ ∆ n ) then � (Γ ⊢ ∆) Thus, � (Γ ⊢ ∆) . André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 9 / 23

  35. Soundness of dL Theorem dL sequent calculus is sound: every dL sequent with a proof is valid. Proof (by induction on structure of sequent calculus proof). Proofs without rule uses only prove dL axioms, which are sound. 0 Sequent proof ends with some proof step: 1 Γ 1 ⊢ ∆ 1 Γ n ⊢ ∆ n ... Γ ⊢ ∆ The subproof of each premise Γ i ⊢ ∆ i is smaller, so � Γ i ⊢ ∆ i by IH. All dL proof rules are proved sound, also the one used above, i.e.: If � (Γ 1 ⊢ ∆ 1 ) and ... and � (Γ n ⊢ ∆ n ) then � (Γ ⊢ ∆) Thus, � (Γ ⊢ ∆) . Todo Always make sure every axiom and proof rule we adopt is sound! André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 9 / 23

  36. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  37. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ∪ β ] P , ∆ [ ∪ ] L Γ , [ α ∪ β ] P ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  38. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Γ ⊢ [ α ∪ β ] P , ∆ [ ∪ ] L Γ , [ α ∪ β ] P ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  39. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Γ ⊢ [ α ∪ β ] P , ∆ [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ Γ , [ α ∪ β ] P ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  40. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Boring! Already follow from the axiom [ ∪ ] [ α ∪ β ] P ↔ [ α ] P ∧ [ β ] P Γ ⊢ [ α ∪ β ] P , ∆ [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ Γ , [ α ∪ β ] P ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  41. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Boring! Already follow from the axiom [ ∪ ] [ α ∪ β ] P ↔ [ α ] P ∧ [ β ] P Γ ⊢ [ α ∪ β ] P , ∆ Rules [ ∪ ] R , [ ∪ ] L would only apply top-level, [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ not in any other logical context such as [ x ′′ = − g ] _ Γ , [ α ∪ β ] P ⊢ ∆ [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 10 / 23

  42. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Boring! Already follow from the axiom [ ∪ ] [ α ∪ β ] P ↔ [ α ] P ∧ [ β ] P Γ ⊢ [ α ∪ β ] P , ∆ Rules [ ∪ ] R , [ ∪ ] L would only apply top-level, [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ not in any other logical context such as [ x ′′ = − g ] _ Γ , [ α ∪ β ] P ⊢ ∆ Contextual Equivalence: substituting equals for equals CER Γ ⊢ C ( Q ) , ∆ ⊢ P ↔ Q CEL Γ , C ( Q ) ⊢ ∆ ⊢ P ↔ Q Γ ⊢ C ( P ) , ∆ Γ , C ( P ) ⊢ ∆ [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 11 / 23

  43. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Boring! Already follow from the axiom [ ∪ ] [ α ∪ β ] P ↔ [ α ] P ∧ [ β ] P Γ ⊢ [ α ∪ β ] P , ∆ Rules [ ∪ ] R , [ ∪ ] L would only apply top-level, [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ not in any other logical context such as [ x ′′ = − g ] _ Γ , [ α ∪ β ] P ⊢ ∆ Contextual Equivalence: substituting equals for equals CER Γ ⊢ C ( Q ) , ∆ ⊢ P ↔ Q CEL Γ , C ( Q ) ⊢ ∆ ⊢ P ↔ Q Γ ⊢ C ( P ) , ∆ Γ , C ( P ) ⊢ ∆ [? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) ↔ [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 11 / 23

  44. Dynamics Proof Rules of Sequent Calculus Have: Left and right proof rule for all propositional connectives Need: Left and right proof rule for all top-level operators in all modalities? [ ∪ ] R Γ ⊢ [ α ] P ∧ [ β ] P , ∆ Boring! Already follow from the axiom [ ∪ ] [ α ∪ β ] P ↔ [ α ] P ∧ [ β ] P Γ ⊢ [ α ∪ β ] P , ∆ Rules [ ∪ ] R , [ ∪ ] L would only apply top-level, [ ∪ ] L Γ , [ α ] P ∧ [ β ] P ⊢ ∆ not in any other logical context such as [ x ′′ = − g ] _ Γ , [ α ∪ β ] P ⊢ ∆ Contextual Equivalence: substituting equals for equals CER Γ ⊢ C ( Q ) , ∆ ⊢ P ↔ Q CEL Γ , C ( Q ) ⊢ ∆ ⊢ P ↔ Q Γ ⊢ C ( P ) , ∆ Γ , C ( P ) ⊢ ∆ [? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) ↔ [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 11 / 23

  45. Simple Example Proof Dynamics in Sequent Calculus [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  46. Simple Example Proof Dynamics in Sequent Calculus � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [ a := − b ; c := 10 ] ↔ � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [ a := − b ][ c := 10 ] by [;] [:=] ⊢ [ a := − b ][ c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  47. Simple Example Proof Dynamics in Sequent Calculus � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [ a := − b ][ c := 10 ] ↔ � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [ c := 10 ] by [:=] [:=] ⊢ [ c := 10 ] � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [:=] ⊢ [ a := − b ][ c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  48. Simple Example Proof Dynamics in Sequent Calculus � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [ c := 10 ] ↔ v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) by [:=] ⊢ v 2 ≤ 10 ∧ − ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) [:=] ⊢ [ c := 10 ] � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [:=] ⊢ [ a := − b ][ c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  49. Simple Example Proof Dynamics in Sequent Calculus ∗ id v 2 ≤ 10 , b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 ∗ id v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∨ R v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) ⊢ v 2 ≤ 10 ∧ − ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) [:=] ⊢ [ c := 10 ] � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [:=] ⊢ [ a := − b ][ c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  50. Simple Example Proof Dynamics in Sequent Calculus ∗ id v 2 ≤ 10 , b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 ∗ id v 2 ≤ 10 , b > 0 ⊢ b > 0 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) , v 2 ≤ 10 ∧ L v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∨ R v 2 ≤ 10 ∧ b > 0 ⊢ ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ∧ R v 2 ≤ 10 ∧ b > 0 ⊢ b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) → R ⊢ v 2 ≤ 10 ∧ b > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) ⊢ v 2 ≤ 10 ∧ − ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ 10 ) [:=] ⊢ [ c := 10 ] � � v 2 ≤ 10 ∧− ( − b ) > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [:=] ⊢ [ a := − b ][ c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) [;] ⊢ [ a := − b ; c := 10 ] � � v 2 ≤ 10 ∧− a > 0 → b > 0 ∧ ( ¬ ( v ≥ 0 ) ∨ v 2 ≤ c ) Need to reason about real arithmetic Here: to glue previous propositional proof with this dynamic proof André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 12 / 23

  51. Quantifier Proof Rules ∀ R Γ ⊢ ∀ x p ( x ) , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  52. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R Γ ⊢ ∀ x p ( x ) , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  53. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ ∀ R: show for fresh variable y about which we can’t know anything André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  54. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  55. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  56. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) ∃ R: enough to show for any witness term e André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  57. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ ∀ L Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  58. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  59. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L ( arbitrary term e ) Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) ∀ L: even holds for arbitrary term e André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  60. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L ( arbitrary term e ) Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) ∃ L Γ , ∃ x p ( x ) ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  61. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L ( arbitrary term e ) Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) Γ , p ( y ) ⊢ ∆ ∃ L Γ , ∃ x p ( x ) ⊢ ∆ André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  62. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L ( arbitrary term e ) Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) Γ , p ( y ) ⊢ ∆ ∃ L ( y �∈ Γ , ∆ , ∃ x p ( x )) Γ , ∃ x p ( x ) ⊢ ∆ ∃ L: assume for fresh variable y about which we can’t know anything André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  63. Quantifier Proof Rules Γ ⊢ p ( y ) , ∆ ∀ R ( y �∈ Γ , ∆ , ∀ x p ( x )) Γ ⊢ ∀ x p ( x ) , ∆ Γ , p ( e ) ⊢ ∆ ∀ L ( arbitrary term e ) Γ , ∀ x p ( x ) ⊢ ∆ Γ ⊢ p ( e ) , ∆ ∃ R Γ ⊢ ∃ x p ( x ) , ∆ ( arbitrary term e ) Γ , p ( y ) ⊢ ∆ ∃ L ( y �∈ Γ , ∆ , ∃ x p ( x )) Γ , ∃ x p ( x ) ⊢ ∆ Important: soundness means that conclusion valid if all premises valid. André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 13 / 23

  64. A Sequent Proof of a Single-hop Bouncing Ball ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 14 / 23

  65. A Sequent Proof of a Single-hop Bouncing Ball [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 14 / 23

  66. A Sequent Proof of a Single-hop Bouncing Ball � ( H − g 2 t 2 = 0 → B ( H − g 2 t 2 , − c ( − gt ))) ∧ ( H − g 2 t 2 ≥ 0 → B ( H − g 2 t 2 , − gt A ⊢ ∀ t ≥ 0 � � [:=] A ⊢ ∀ t ≥ 0 [ x := H − g 2 t 2 ] ( x = 0 → B ( x , − c ( − gt ))) ∧ ( x ≥ 0 → B ( x , − gt )) � � [:=] A ⊢ ∀ t ≥ 0 [ x := H − g 2 t 2 ][ v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [;] A ⊢ ∀ t ≥ 0 [ x := H − g � � 2 t 2 ; v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [ ′ ] A ⊢ [ x ′′ = − g ] � � ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 14 / 23

  67. A Sequent Proof of a Single-hop Bouncing Ball [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ↔ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) by [;] [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  68. A Sequent Proof of a Single-hop Bouncing Ball [? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) ↔ � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) by [ ∪ ] [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  69. A Sequent Proof of a Single-hop Bouncing Ball [? x = 0 ; v := − cv ] B ( x , v ) ↔ [? x = 0 ][ v := − cv ] B ( x , v ) by [;] [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  70. A Sequent Proof of a Single-hop Bouncing Ball [? x = 0 ][ v := − cv ] B ( x , v ) ↔ x = 0 → [ v := − cv ] B ( x , v ) by [?] � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  71. A Sequent Proof of a Single-hop Bouncing Ball [ v := − cv ] B ( x , v ) ↔ x = 0 → B ( x , − cv ) by [:=] [ ′ ] A ⊢ [ x ′′ = − g ] � � ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  72. A Sequent Proof of a Single-hop Bouncing Ball [ ′ ] [ x ′ = f ( x )] p ( x ) ↔ ∀ t ≥ 0 [ x := y ( t )] p ( x ) [;] A ⊢ ∀ t ≥ 0 [ x := H − g � � 2 t 2 ; v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [ ′ ] A ⊢ [ x ′′ = − g ] � � ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  73. A Sequent Proof of a Single-hop Bouncing Ball � � [:=] A ⊢ ∀ t ≥ 0 [ x := H − g 2 t 2 ][ v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [;] A ⊢ ∀ t ≥ 0 [ x := H − g � � 2 t 2 ; v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [ ′ ] A ⊢ [ x ′′ = − g ] � � ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

  74. A Sequent Proof of a Single-hop Bouncing Ball � � [:=] A ⊢ ∀ t ≥ 0 [ x := H − g 2 t 2 ] ( x = 0 → B ( x , − c ( − gt ))) ∧ ( x ≥ 0 → B ( x , − gt )) � � [:=] A ⊢ ∀ t ≥ 0 [ x := H − g 2 t 2 ][ v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [;] A ⊢ ∀ t ≥ 0 [ x := H − g � � 2 t 2 ; v := − gt ] ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) [ ′ ] A ⊢ [ x ′′ = − g ] � � ( x = 0 → B ( x , − cv )) ∧ ( x ≥ 0 → B ( x , v )) � � [:=] A ⊢ [ x ′′ = − g ] ( x = 0 → [ v := − cv ] B ( x , v )) ∧ ( x ≥ 0 → B ( x , v )) [?] A ⊢ [ x ′′ = − g ] � � [? x = 0 ][ v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ] � � [? x = 0 ; v := − cv ] B ( x , v ) ∧ [? x ≥ 0 ] B ( x , v ) [ ∪ ] A ⊢ [ x ′′ = − g ][? x = 0 ; v := − cv ∪ ? x ≥ 0 ] B ( x , v ) [;] A ⊢ [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) ⊢ A → [ x ′′ = − g ;(? x = 0 ; v := − cv ∪ ? x ≥ 0 )] B ( x , v ) → R def A ≡ 0 ≤ x ∧ x = H ∧ v = 0 ∧ g > 0 ∧ 1 ≥ c ≥ 0 def B ( x , v ) ≡ 0 ≤ x ∧ x ≤ H { x ′′ = − g } def ≡ { x ′ = v , v ′ = − g } André Platzer (CMU) LFCPS/06: Truth & Proof LFCPS/06 15 / 23

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Focusing for d L 15-824 Logical Foundations of Cyber-Physical Systems Fall 2018 Klaas Pruiksma

Focusing for d L 15-824 Logical Foundations of Cyber-Physical Systems Fall 2018 Klaas Pruiksma

Focusing for d L 15-824 Logical Foundations of Cyber-Physical Systems Fall 2018 Klaas Pruiksma December 10, 2018 1 / 10 Goal Develop a focused version of Differential Dynamic Logic(d L ) [3], with the intent that it serve as a basis for

260 views • 10 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/09:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/09:

09: Reactions & Delays Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 1 / 17 Outline Learning

799 views • 59 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/08:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/08:

08: Events & Responses Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/08: Events & Responses LFCPS/08 1 / 20 Outline Learning

719 views • 56 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/04:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/04:

04: Safety & Contracts Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/04: Safety & Contracts LFCPS/04 1 / 16 Outline Learning

1.12k views • 80 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/01:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/01:

Logical Foundations of Cyber-Physical Systems 01: Cyber-Physical Systems: Overview Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/01: Overview LFCPS/01 1 / 28 Outline CPS:

934 views • 49 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/21:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/21:

21: Virtual Substitution & Real Arithmetic Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/21: Virtual Substitution & Real

865 views • 84 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/13:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/13:

13: Differential Invariants & Proof Theory Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/13: Differential Invariants & Proof

1.53k views • 127 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/14:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/14:

14: Hybrid Systems & Games Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 1 / 24 Outline

1.07k views • 102 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/10:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/10:

10: Differential Equations & Differential Invariants Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/10: Differential Equations &

1.02k views • 86 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/20:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/20:

20: Virtual Substitution & Real Equations Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/20: Virtual Substitution & Real Equations

3.49k views • 114 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/17:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/17:

17: Game Proofs & Separations Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/17: Game Proofs & Separations LFCPS/17 1 / 25

1.26k views • 96 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/02:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/02:

02: Differential Equations & Domains Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/02: Differential Equations & Domains LFCPS/02

788 views • 78 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/11:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/11:

11: Differential Equations & Proofs Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11

1.49k views • 111 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/07:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/07:

07: Control Loops & Invariants Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/07: Control Loops & Invariants LFCPS/07 1 / 16

1.45k views • 118 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/15:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/15:

15: Winning Strategies & Regions Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 1 / 23

1.02k views • 101 slides
Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/18:

Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/18:

18: Axioms & Uniform Substitutions Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/18: Axioms & Uniform Substitutions LFCPS/18 1

1.96k views • 170 slides
Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

W HAT YOU WILL LEARN how to use a theorem prover background, how it works how to prove and specify NICTA Advanced Course Slide 1 Slide 3 Theorem Proving Health Warning Principles, Techniques, Applications Theorem Proving is

262 views • 9 slides
Soundness in negotiations Igor Walukiewicz CNRS, Bordeaux University Joint work with Javier

Soundness in negotiations Igor Walukiewicz CNRS, Bordeaux University Joint work with Javier

Soundness in negotiations Igor Walukiewicz CNRS, Bordeaux University Joint work with Javier Esparza, Denis Kuperberg, and Anca Muscholl 1 / 25 Verification of concurrent systems suffers from the state explosion problem. 2 / 25 Verification

430 views • 25 slides
A Sound Type System for Secure Flow Analysis Dennis Volpano, Geoffrey Smith, Cynthia Irvine

A Sound Type System for Secure Flow Analysis Dennis Volpano, Geoffrey Smith, Cynthia Irvine

A Sound Type System for Secure Flow Analysis Dennis Volpano, Geoffrey Smith, Cynthia Irvine Presenter: Lantian Zheng CS 711 September 29, 2003 Soundness of Denings Program Certification Mechanism Define the soundness property: S ( P )

819 views • 24 slides
Worst-case to average case reductions for the distance to a code CCC 2018 Eli Ben-Sasson and

Worst-case to average case reductions for the distance to a code CCC 2018 Eli Ben-Sasson and

Worst-case to average case reductions for the distance to a code CCC 2018 Eli Ben-Sasson and Swastik Kopparty and Shubhangi Saraf June 2018 Overview motivation main results applications one proof Motivation Arithmetization

283 views • 15 slides
Logic for real number computation Helmut Schwichtenberg (j.w.w. Ulrich Berger, Kenji Miyamoto and

Logic for real number computation Helmut Schwichtenberg (j.w.w. Ulrich Berger, Kenji Miyamoto and

Logic for real number computation Helmut Schwichtenberg (j.w.w. Ulrich Berger, Kenji Miyamoto and Hideki Tsuiki) Mathematisches Institut, LMU, M unchen Trends in Proof Theory, Hamburg, 20. - 21. September 2015 1 / 16 Dyadic rationals: a i

337 views • 16 slides
Supermartingales for Reachability in in Probabilistic Programs Toru Takisaka 1 , Yuichiro Oyabu

Supermartingales for Reachability in in Probabilistic Programs Toru Takisaka 1 , Yuichiro Oyabu

Ranking and Repulsing Supermartingales for Reachability in in Probabilistic Programs Toru Takisaka 1 , Yuichiro Oyabu 2,3 , Natsuki Urabe 1 , Ichiro Hasuo 2,3 National Institute of Informatics, Japan 1 The Graduate University for Advanced

748 views • 62 slides
Symbolic Encryption with Pseudorandom Keys Daniele Micciancio (UCSD) Cryptographic Protocols

Symbolic Encryption with Pseudorandom Keys Daniele Micciancio (UCSD) Cryptographic Protocols

Symbolic Encryption with Pseudorandom Keys Daniele Micciancio (UCSD) Cryptographic Protocols Often make use of several basic crypto ops: (Enc) Encryption: E(k,m) (PRG) Pseudorandom generators: G(k) Protocol should be secure for

433 views • 24 slides
Proof nets for sumproduct logic Willem Heijltjes LFCS School of Informatics University of

Proof nets for sumproduct logic Willem Heijltjes LFCS School of Informatics University of

Proof nets for sumproduct logic Willem Heijltjes LFCS School of Informatics University of Edinburgh Kananaskis, 11-12 June 2011 In part 2 Recap The soundness proof The full net calculus a A B 0 1 f f 0 + + g 1 f f

1.51k views • 112 slides

More recommend