Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication
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Definition
- $\paren {p \land q} \lor \paren {p \land r} \vdash p \land \paren {q \lor r}$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\paren {p \land q} \lor \paren {p \land r}$ | Premise | (None) | ||
2 | 2 | $p \land q$ | Assumption | (None) | ||
3 | 2 | $p$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
4 | 4 | $p \land r$ | Assumption | (None) | ||
5 | 4 | $p$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
6 | 1 | $p$ | Proof by Cases: $\text{PBC}$ | 1, 2 – 3, 4 – 5 | Assumptions 2 and 4 have been discharged | |
7 | 7 | $p \land q$ | Assumption | (None) | ||
8 | 7 | $q$ | Rule of Simplification: $\land \EE_2$ | 7 | ||
9 | 7 | $q \lor r$ | Rule of Addition: $\lor \II_1$ | 7 | ||
10 | 10 | $p \land r$ | Assumption | (None) | ||
11 | 10 | $r$ | Rule of Simplification: $\land \EE_2$ | 10 | ||
12 | 10 | $q \lor r$ | Rule of Addition: $\lor \II_2$ | 11 | ||
13 | 1 | $q \lor r$ | Proof by Cases: $\text{PBC}$ | 1, 7 – 9, 10 – 12 | Assumptions 7 and 10 have been discharged | |
14 | 1 | $p \land \paren {q \lor r}$ | Rule of Conjunction: $\land \II$ | 6, 13 |
$\blacksquare$
Sources
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercises $1.4: \ 2 \ \text{(q)}$