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2 Magmas algebras and trees
3 Program schemes
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abstract data types algebraic class algebraic semantics algorithm associated belongs binary relation C-free C-Herbrand interpretation characterization class of interpretations class of valuated clearly closed first-order formulas compact elements complete F-magma computation rule computation sequence construction continuous Corollary data types deduce Definition denotational semantics directed set directed subset discrete interpretations equational classes example exists f in F F-algebra F-magma preorder finite formal function computed function variables hence Herbrand implies infinite tree integer interpretation defined intuitively languages least fixpoint least upper bound Let F Let t,t macroexpansions magma monotone morphism node notation obtained occurrences of function Operational semantics ordered F-magma ordered set otherwise outermost program scheme program transformations proofs of program proposition 5.25 prove recursive scheme relation on M(F resp result rewriting system RPS's substitution substitution-closed syntactic theorem tions trees on FuV ultraproducts useless valuated interpretation vector whence