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**1 - 2**of**2**### Extending the Type Checker of SML by Polymorphic Recursion: A Correctness Proof

, 1997

"... . We describe an extension of the type inference of Standard ML that covers polymorphic recursion. For any term t of SML, a type scheme ø and a system L of inequations between (simple) types is computed, such that the types of t are the instances of ø by substitutions S that satisfy L. The inequat ..."

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. We describe an extension of the type inference of Standard ML that covers polymorphic recursion. For any term t of SML, a type scheme ø and a system L of inequations between (simple) types is computed, such that the types of t are the instances of ø by substitutions S that satisfy L. The inequation constraints L are computed bottom-up in a modification of Milner's algorithm W . The correctness proof is complicated by the fact that unknowns for polytypes are needed -- in contrast to type inference for SML. 1 Introduction Functional programming languages like ML[19], Miranda[23], or Haskell[?], have made statically typed polymorphic languages popular. Their success depends to a large extend on the following properties of the underlying type system of Damas/Milner[2]: - typability of an untyped term is decidable, - for typable terms, a schema representing the set of its types can be inferred automatically, - the declaration of polymorphic values by the user is supported, - well-typed...

### Simultaneous Rigid E-Unification is not so Simple

, 1995

"... Simultaneous rigid E-unification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid E-unification. There ..."

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Simultaneous rigid E-unification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid E-unification. There were several faulty proofs of the decidability of this problem. In this article we prove several results about the simultaneous rigid E-unification. Two results are reductions of known problems to simultaneous rigid E-unification. Both these problems are very hard. The word equation solving (unification under associativity) is reduced to the monadic case of simultaneous rigid E-unification. The variable-bounded semi-unification problem is reduced to the general simultaneous rigid E-unification. The word equation problem used in the first reduction is known to be decidable, but the decidability result is extremely non-trivial. As for the variablebounded semi-unification, its decidability is ...