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| For more details, see [[http://verify.stanford.edu/summerschool2004|Combination of Decision Procedures Summer School 2004]]. | For more details, see [[http://verify.stanford.edu/summerschool2004|Combination of Decision Procedures Summer School 2004]]. | ||
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| ===== The notion of formal proof ===== | ===== The notion of formal proof ===== | ||
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| Notion of formal proof. | Notion of formal proof. | ||
| - | How do we define a prove in a program? | + | How do we define a proof in a program? |
| - | * Set of axiomes: Ax_1, Ax_2, ..., Ax_n | + | * Set of axioms: Ax_1, Ax_2, ..., Ax_n |
| * Set of inference rules | * Set of inference rules | ||
| - | A prouve then is nothing more than a sequence of formulas F_i. | + | A proof then is nothing more than a sequence of formulas F_i. |
| - | __Informal Definition:__ A prove is such that each F_i can be derived either from an axiom or | + | __Informal Definition:__ A proof is such that each F_i can be derived either from an axiom or |
| from a rule or from a formula F_j (j < i) that we have previously derived. | from a rule or from a formula F_j (j < i) that we have previously derived. | ||
| Line 189: | Line 188: | ||
| Proof rules for first-order logic. | Proof rules for first-order logic. | ||
| * propositional axioms: an instance of propositional tautology is an axiom | * propositional axioms: an instance of propositional tautology is an axiom | ||
| - | * instantiation: from (ALL x.F) derive F[x:=t]. | + | * instantiation: from (∀x.F) derive F[x:=t]. |
| - | * generalization: after proving F which contains a fresh variable x, conclude (ALL x.F) | + | * generalization: after proving F which contains a fresh variable x, conclude (∀x.F) |
| Example: axiomatizing some set operations. | Example: axiomatizing some set operations. | ||