Concrete Semantics: With Isabelle/HOL by Tobias Nipkow, Gerwin Klein

By Tobias Nipkow, Gerwin Klein

Part I of this publication is a pragmatic creation to operating with the Isabelle evidence assistant. It teaches you ways to write down useful courses and inductive definitions and the way to end up homes approximately them in Isabelle’s based facts language. half II is an creation to the semantics of relevant languages with an emphasis on functions like compilers and software analysers. The distinguishing characteristic is that each one the math has been formalised in Isabelle and masses of it truly is executable. half I focusses at the info of proofs in Isabelle; half II might be learn even with no familiarity with Isabelle’s evidence language, all proofs are defined intimately yet informally.

The ebook teaches the reader the paintings of designated logical reasoning and the sensible use of an evidence assistant as a surgical device for formal proofs approximately computing device technology artefacts. during this experience it represents a proper method of laptop technological know-how, not only semantics. The Isabelle formalisation, together with the proofs and accompanying slides, are freely to be had on-line, and the publication is acceptable for graduate scholars, complicated undergraduate scholars, and researchers in theoretical desktop technological know-how and logic.

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Example text

Isabelle knows about these and a number of other introduction rules. The command apply rule automatically selects the appropriate rule for the current subgoal. You can also turn your own theorems into introduction rules by giving them the intro attribute, analogous to the simp attribute. In that case blast, fastforce and (to a limited extent) auto will automatically backchain with those theorems. The intro attribute should be used with care because it increases the search space and can lead to nontermination.

Thus we guide auto towards the case distinction we made in our proof above. Let us extend constant folding: Plus (N 0) a and Plus a (N 0) should be replaced by a. 1 Arithmetic Expressions 31 process into two functions: one performs the local optimizations, the other traverses the term. induct). Now we replace Plus by plus in a bottom-up manner throughout an expression: fun asimp :: "aexp ⇒ aexp" where "asimp (N n) = N n" | "asimp (V x ) = V x" | "asimp (Plus a 1 a 2 ) = plus (asimp a 1 ) (asimp a 2 )" Correctness is expressed exactly as for asimp_const : lemma "aval (asimp a) s = aval a s" The proof is by structural induction on a; the Plus case follows with the help of Lemma aval_plus.

For example, le_trans, transitivity of on type nat, is not an introduction rule by default because of the disastrous effect on the search space, but can be useful in specific situations: lemma "[[ (a::nat) b; b by(blast intro: le_trans) c; c d; d e ]] =⇒ a e" Of course this is just an example and could be proved by arith, too. 4 Forward Proof Forward proof means deriving new theorems from old theorems. We have already seen a very simple form of forward proof: the of operator for instantiating unknowns in a theorem.

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