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And he has proofs that these axiomatisations are sound and complete with respect to the described semantics.
The ω rule is an analogue in the λ-calculus of the rule of inference under the same name in formal number theory, according to which one can conclude the universal formula ∀xφ provided one has proofs for φ(x := 0), φ(x := 1), ….
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Finally, this pudding has proof.
Chad has proofed several loaves of his big country bread.
Parker has proof of posting his own version.
To what extent has proof theory achieved its original aims?
This lady has proof of residence.
Paul says UFOs exist, and he has proof.
BS has proof read and planned the study.
"We have proof".
You have proof.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com