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sav08:proofs_and_induction [2008/02/20 22:28] vkuncak |
sav08:proofs_and_induction [2008/02/21 16:49] vkuncak |
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====== Informal Proofs and Mathematical Induction ====== | ====== Informal Proofs and Mathematical Induction ====== | ||
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===== Proof Rules ===== | ===== Proof Rules ===== | ||
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Instead of proving a goal, you can assume its negation and prove any remaining goals (this generalizes proof by contradiction). | Instead of proving a goal, you can assume its negation and prove any remaining goals (this generalizes proof by contradiction). | ||
\[ | \[ | ||
- | \frac{\lnot P \vdash\} | + | \frac{\lnot P \vdash} |
{\vdash P} | {\vdash P} | ||
\] | \] | ||
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When you have an existentially quantified assumption, you can pick a witness for existential statement and denote it by fresh variable "x_0". Informally we would say "let $x_0$ be such $x$". | When you have an existentially quantified assumption, you can pick a witness for existential statement and denote it by fresh variable "x_0". Informally we would say "let $x_0$ be such $x$". | ||
\[ | \[ | ||
- | \frac{P(x_0)} | + | \frac{P(x_0) \vdash} |
{\exists x. P(x) \vdash} | {\exists x. P(x) \vdash} | ||
\] | \] | ||
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===== Using Proof Rules in Isabelle ===== | ===== Using Proof Rules in Isabelle ===== | ||
+ | |||
+ | [[Isabelle theorem prover]] implements these rules. It names them: | ||
+ | * conjE, conjI, disjE, disjI1, disjI2, impE, impI, allE, allI, exE, exI | ||
+ | |||
+ | The "E" (elimination) rules are for constructs on left-hand side, and "I" (introduction) rules are for constructs on right-hand side. | ||
+ | |||
+ | More information: [[http://isabelle.in.tum.de/dist/Isabelle/doc/tutorial.pdf|Isabelle tutorial]], Chapter 5 "The Rules of the Game". | ||
+ | |||
+ | Demo: Proving above example in Isabelle. | ||
===== Completeness for First-Order Logic ===== | ===== Completeness for First-Order Logic ===== |