# Jack and Jill

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| $J_a \cap J_i$ | $\frac{dh}{dt} > 0$ | $(h_{base} \to h_{well}) \in \mathbb{X}_{hill}$ |
|:--:|:--:|:--:|
| *Jack and Jill* | *went up* | *the hill,* |

$$J_{a}, J_{i} \in \{\text{kids(h, p, c)}\}; \space \frac{dh}{dt} > 0$$

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| $(x_{i}, \emptyset) \to (x_{j}, \mathbb{1}) \to (x_{i},\mathbb{1})$ | $p \ni w$ | $= p_{full}$ |
|:--:|:--:|:--:|
| *To fetch* | *a pail of* | *water.* |

$$(p_{full} \cap h_{well}) \to (p_{full} \cap h_{base})$$

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| $J_a$ | $\frac{dh_{J_{a}}}{dt} \bigg\|_{t=t_1} \ll 0$ |
|:--:|:--:|
| *Jack* | *fell down,* |

$$\exists t_1 \text{ s.t. } \frac{dh_{J_{a}}}{dt} \bigg|_{t=t_1} \ll 0$$

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| $c\{c_{1}\} \to c\{\bigcup_{i=1}^{n>1} c_i\}$ | $c_{J_{a}}$ |
|:--:|:--:|
| *And broke* | *his crown.* |

$$c_{1} \to \bigcup_{i=1}^{n} c_i \quad (n \ge 2, \quad C_i \cap C_j = \emptyset \text{ for } i \neq j)$$

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| $J_{i}$ | $\frac{dh_{J_{i}}}{dt} \bigg\|_{t=t_2} \ll 0$ | $\exists t_2 > t_1$ |
|:--:|:--:|:--:|
| *And Jill* | *came tumbling* | *after.* |

$$\exists t_2 > t_1 \text{ s.t. } \frac{dh_{J_{i}}}{dt} \bigg|_{t=t_2} \ll 0$$

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