By N. Shankar
The automated verification of enormous elements of arithmetic has been an objective of many mathematicians from Leibniz to Hilbert. whereas G?del's first incompleteness theorem confirmed that no desktop application might instantly turn out yes actual theorems in arithmetic, the arrival of digital pcs and complex software program potential in perform there are various fairly powerful structures for automatic reasoning that may be used for checking mathematical proofs. This ebook describes using a working laptop or computer application to examine the proofs of numerous celebrated theorems in metamathematics together with these of G?del and Church-Rosser. the pc verification utilizing the Boyer-Moore theorem prover yields certain and rigorous proofs of those tough theorems. It additionally demonstrates the variety and tool of computerized facts checking know-how. The mechanization of metamathematics itself has vital implications for computerized reasoning, simply because metatheorems might be utilized as labor-saving units to simplify evidence building.
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Extra info for Metamathematics, machines and Goedel's proof
Starting from the bigstep interpreter, in each phase, the case analysis is lifted out so that the interpreter becomes a set of one-argument functions on the active expression that produce a function accepting the passive expression as an extra parameter. M. Ariola et al. c e t = λτ. c t E = c c [E/α] V t E=E V α E V =αV E V = λτ. λτ. t μ ˜[x]. V F τ τ [x = t] τ V F = λτ. E V ]τ )) τ Fig. 2. Continuation and environment passing style transform for λ[lvτ ∗] interpreter as they become available. The partial evaluation of the interpreter with only the active argument becomes a continuation waiting for the passive counterpart.
Theorem 6. If cτ [lv∗] c τ then c τ =βη c τ . Store-Based CPS. Another approach to deriving a CPS for a classical callby-need calculus is to extend the CPS given by Okasaki et al.  with control operators. We do so by adding the operators μ and “bracket” from Parigot’s λμcalculus  for respectively capturing and re-installing an evaluation context. This extension corresponds to applying a store passing transformation to the transformation given in Figure 3. The natural extension of the delay and force CPS results in a diﬀerent semantics than the one in Figure 2.
X||α )||α ||˜ μ[a]. a||fst · μ ˜[f ]. f ||q · (I, I) · tp [q = μδ. a||snd · δ ][f = μβ. a||fst · β ][a = μα. μ . μ . x||α )||α [q = μδ. a||snd · δ ][f = μβ. a||fst · β ][a = μα. μ . x||α )||α ] where α = μ ˜ [a]. a||fst · μ ˜[f ]. f ||q · (I, I) · tp → → q||(I, I) · tp [q = μδ. μ . μ . μ . μ . μ . x||α ][f = I][a = (I, I)] where α = μ ˜ [a]. a||fst · μ ˜[f ]. f ||q · (I, I) · tp →... M. Ariola et al. We saw that using the multicut calculus λ[lv τ ] , the program produces (I, I)||tp as a result — every time the continuation α is invoked, both f and q are reverted to their unevaluated states.