In derivations of a sequent system, |$\mathcal{L}\mathcal{J}$|⁠, and a natural deduction system, |$\mathcal{N}\mathcal{J}$|⁠, the trails of formulae and the subformula property based on these trails will be defined. The derivations of |$\mathcal{N}\mathcal{J}$| and |$\mathcal{L}\mathcal{J}$| will be connected by the map |$g$|⁠, and it will be proved the following: an |$\mathcal{N}\mathcal{J}$|-derivation is normal |$\Longleftrightarrow$|⁠, it has the subformula property based on trails |$\Longleftrightarrow$|⁠, its |$g$|-image in |$\mathcal{L}\mathcal{J}$| is without maximum cuts |$\Longrightarrow$| that |$g$|-image has the subformula property based on trails. In |$\mathcal{L}\mathcal{J}$|-derivations, another type of cuts, sub-cuts, will be introduced, and it will be proved the following: all cuts of an |$\mathcal{L}\mathcal{J}$|-derivation are sub-cuts |$\Longleftrightarrow$|⁠, it has the subformula property based on trails.

中文翻译：

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自然推导和解析割的子公式性质

在派生系统| $ \ mathcal {L} \ mathcal {J} $ |⁠和自然演绎系统| $ \ mathcal {N} \ mathcal {J} $ |⁠的派生中，公式和将定义基于这些轨迹的子公式属性。的推导| $ \ mathcal {N} \ {mathcal}Ĵ$ | 和| $ \ mathcal {L} \ mathcal {J} $ | 将由地图| $ g $ |⁠连接，并且将证明以下内容：| $ \ mathcal {N} \ mathcal {J} $ | -派生是正常的| $ \ Longleftrightarrow $ |⁠，它具有基于路径的子公式属性| $ \ Longleftrightarrow $ |⁠，其| $ g $ | - $ | $ {mathcal {L} \ mathcal {J} $ |中的图像 没有最大的削减| $ \ Longrightarrow $ | 那| $ g $ | -image具有基于轨迹的子公式属性。在| $ \ mathcal {L} \ mathcal {J} $ |中 -派生，另一种削减形式，即子削减，将被证明如下：| $ \ mathcal {L} \ mathcal {J} $ |的所有削减 -derivation是子剪切| $ \ Longleftrightarrow $ |⁠，它具有基于轨迹的subformula属性。