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If the following SLD resolution steps are possible

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(1)

Exchangement Lemma (Lemma 4.3.2)

If the following SLD resolution steps are possible

{¬A1, . . . ,¬Ai, . . . ,¬Aj, . . . ,¬Ak} {B,¬C1, . . . ,¬Cn} σ1({¬A1, . . . ,¬C1, . . . ,¬Cn, . . . ,¬Aj, . . . ,¬Ak}) {D,¬E1, . . . ,¬Em} σ2(σ1({¬A1, . . . ,¬C1, . . . ,¬Cn, . . . ,¬E1, . . . ,¬Em, . . . ,¬Ak}))

then the following SLD resolution steps are possible as well:

{¬A1, . . . ,¬Ai, . . . ,¬Aj, . . . ,¬Ak} {D,¬E1, . . . ,¬Em} σ1′({¬A1, . . . ,¬Ai, . . . , E1, . . . ,¬Em . . . ,¬Ak}) {B,¬C1, . . . ,¬Cn} σ2′ (σ1′({¬A1, . . . ,¬C1, . . . ,¬Cn, . . . ,¬E1, . . . ,¬Em, . . . ,¬Ak}))

Here, σ

2′

◦ σ

1′

= ν ◦ σ

2

◦ σ

1

for a variable renaming ν .

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