General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

The deduction theorem and the entailment theorem together allow us to understand the claim that
the two formulations of soundness (A ⊢ B implies A |= B and ⊢ A implies |=B) are equivalent.
Indeed, if we have A ⊢ B, then by the deduction theorem ⊢ A ⇒ B, and thus |=A ⇒ B by
soundness, which gives us A |= B by the entailment theorem. The other direction and the
argument for the corresponding statement about completeness are similar.
Of course this is still not the version of the proof we originally wanted, since it talks about the
Hilbert Calculus H 1 , but we can show that H 1 and H 0 are equivalent.
But as we will see, the derivability relations induced by the two caluli are the same. So we can
prove the original theorem after all.
The Deduction Theorem for H 0
Lemma 324 ⊢ H 1 = ⊢ H 0
Proof:
P.1 All H 1 axioms are H 0 theorems. (by Subst)
P.2 For the other direction, we need a proof transformation argument:
P.3 We can replace an application of MP followed by Subst by two Subst applications followed
by one MP.
P.4 . . . A ⇒ B . . . A . . . B . . . [C/X](B) . . . is replaced by
. . . A ⇒ B . . . [C/X](A) ⇒ [C/X](B) . . . A . . . [C/X](A) . . . [C/X](B) . . .
P.5 Thus we can push later Subst applications to the axioms, transforming a H 0 proof into
a H 1 proof.
Corollary 325 H, A ⊢ H 0 B, iff H ⊢ H 0 A ⇒ B.
Proof Sketch: by MP and ⊢ H 1 = ⊢ H 0
c○: Michael Kohlhase 193
We can now collect all the pieces and give the full statement of the soundness theorem for H 0
H 0 is sound (full version)
Theorem 326 For all propositions A, B, we have A ⊢ H 0 B implies A |= B.
Proof:
P.1 By deduction theorem A ⊢ H 0 B, iff ⊢ A ⇒ C,
P.2 by the first soundness theorem this is the case, iff |=A ⇒ B,
P.3 by the entailment theorem this holds, iff A |= C.
2.6.5 A Calculus for Mathtalk
c○: Michael Kohlhase 194
In our introduction to Subsection 2.6.0 we have positioned Boolean expressions (and proposition
logic) as a system for understanding the mathematical language “mathtalk” introduced in Subsection
2.2.1. We have been using this language to state properties of objects and prove them all
through this course without making the rules the govern this activity fully explicit. We will rectify
this now: First we give a calculus that tries to mimic the the informal rules mathematicians use
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