Proof by Cases/Sequent Form/Proof 1
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Theorem
Proof by Cases can be symbolised by the sequent:
- $p \lor q, \paren {p \vdash r}, \paren {q \vdash r} \vdash r$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \lor q$ | Premise | (None) | ||
2 | 2 | $p$ | Assumption | (None) | ||
3 | 2 | $r$ | By hypothesis | 2 | as $p \vdash r$ | |
4 | 4 | $q$ | Assumption | (None) | ||
5 | 4 | $r$ | By hypothesis | 4 | as $q \vdash r$ | |
6 | 1 | $r$ | Proof by Cases: $\text{PBC}$ | 1 , 2 – 3 , 4 – 5 | Assumptions 2 and 4 have been discharged |
$\blacksquare$