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sav08:proofs_and_induction [2008/02/21 10:24] 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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If you have a universally quantified assumption, you can instantiate universal quantifier with any value, denoted by term //t// (this is what "for all" means). You keep the original quantified assumption in case you need to instantiate it multiple times. | If you have a universally quantified assumption, you can instantiate universal quantifier with any value, denoted by term //t// (this is what "for all" means). You keep the original quantified assumption in case you need to instantiate it multiple times. | ||
\[ | \[ | ||
- | \frac{P(t), \forall x.P(x)} | + | \frac{P(t), \forall x.P(x) \vdash} |
{\forall x.P(x) \vdash} | {\forall x.P(x) \vdash} | ||
\] | \] | ||
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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 $x$". | ||
\[ | \[ | ||
- | \frac{P(x_0)} | + | \frac{P(x_0) \vdash} |
{\exists x. P(x) \vdash} | {\exists x. P(x) \vdash} | ||
\] | \] |