# Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication

< Rule of Distribution | Conjunction Distributes over Disjunction | Left Distributive | Formulation 1

## Definition

- $\left({p \land q}\right) \lor \left({p \land r}\right) \vdash p \land \left({q \lor r}\right)$

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\left({p \land q}\right) \lor \left({p \land r}\right)$ | Premise | (None) | ||

2 | 2 | $p \land q$ | Assumption | (None) | ||

3 | 2 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 2 | ||

4 | 4 | $p \land r$ | Assumption | (None) | ||

5 | 4 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 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: $\land \mathcal E_2$ | 7 | ||

9 | 7 | $q \lor r$ | Rule of Addition: $\lor \mathcal I_1$ | 7 | ||

10 | 10 | $p \land r$ | Assumption | (None) | ||

11 | 10 | $r$ | Rule of Simplification: $\land \mathcal E_2$ | 10 | ||

12 | 10 | $q \lor r$ | Rule of Addition: $\lor \mathcal I_2$ | 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 | $p \land \left({q \lor r}\right)$ | Rule of Conjunction: $\land \mathcal I$ | 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 $1.4: \ 2 \ \text{(q)}$