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Theorem pbdtr-def

⊦ pbd(βf) ↔ (∀ sf, af, bl. (∀ n. βf(⌜sf.take(n + 1)⌝ , ⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ βf(⌜sf⌝ , ⌜af⌝ ) ⊑ ⌜bl⌝ )