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The Weak normalization theorem for simply typed lambda calculus states that every well-typed term of the simply typed lambda calculus admits at least one finite sequence of beta reductions ending in a beta-normal form.

Suppose a fixed-point combinator $F$ were typable in the simply typed lambda calculus. By the Weak normalization theorem for simply typed lambda calculus, it would have a beta-normal form. For a fresh variable $f$, the term $Ff$ would then also possess a beta-normal form, say $N$.

The fixed-point property givesReducing the occurrence of $Ff$ on the right to $N$ gives the normal form $fN$. The Church-Rosser theorem says that these beta-equivalent terms must have alpha-equivalent normal forms. This is impossible because $fN$ contains more symbols than $N$. Hence no typing context and simple type can type $F$.

The Weak normalization theorem for simply typed lambda calculus says that every term typable in the implicational simply typed lambda calculus has a beta-normal form. Define reducibility by induction on types: a term of atomic type is reducible when it is weakly normalizing, and $M:A\to B$ is reducible when $MN$ is reducible at $B$ for every reducible $N:A$. Induction on types shows that every reducible term is weakly normalizing and that variables are reducible.

The fundamental substitution lemma is proved by induction on a typing derivation: if $\Gamma ⊢M:A$ and each variable in $\Gamma$ is replaced by a reducible term of its declared type, then the resulting term is reducible at $A$. The application case is the definition at arrow type; in the abstraction case, applying the abstraction to any reducible argument makes one beta step to the substituted body, which is reducible by induction. Substituting each free variable by itself makes every well-typed term reducible, hence weakly normalizing.