According to Chris Hankin in his book (Lambda Calculus a Guide for Computer Scientists). A reduction sequence $ \sigma: M_0 \to^{\Delta_0} M_1 \to^{\Delta_1}M_2 \to^{\Delta_2}\ldots $ is a standard reduction if for any pair $ (\Delta_i, \Delta_{i+1})$ , $ \Delta_{i+1}$ is not a residual of a redex to left of $ \Delta_{i}$ relative to the given reduction from $ M_{i}$ to $ M_{i}$ . The Standardization Theorem says that if a term $ A$ $ \beta$ -reduces to $ B$ , then there is a standard path from $ A$ to $ B$ . On the other hand, in the leftmost strategy, outermost redex is always reduced first. The Leftmost Reduction Theorem says that if a term $ A$ $ \beta$ -reduces to $ B$ , and $ B$ is in $ \beta$ -normal form, then the leftmost strategy reaches $ B$ . My question is:

1- I’m not seeing the difference between the reductions Standardization and Leftmost, for me, it seems to say the same thing.

2- Why the Leftmost Reduction Theorem is a particular case of the standardization Theorem?