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Animal sayings are quite common in English, even although we have less countryside than ever. Here are a few interesting and reasonably common ones.
As cunning as the fox. Cunning means clever and skillful in planning to deceive. Synonyms include wily, crafty, and sly. We sometimes say with the cunning of a fox, or as wily as a fox.
This idea has a basis in the fact that the fox can move quietly and stealthily, and can steal and eat domestic poultry, such as hens and ducks. However, it is a stereotype too, because badgers can also move stealthily and steal poultry, but our idea about them is rather different. Badgers are associated with being stubborn
Foxes appear as crafty characters in traditional folk tales and in fairy tales.
As fast as the deer. This saying is less common, but means very fast. You could also say swift.
We sometimes say as fast as a hare.
As brave as the lion. This is a common saying.
Note, you can also say as hungry as a horse. This is quite a different register, though, and whereas the three sayings above would be fine to say in public, as hungry as a horse is very informal.
As slow as a snail. This saying is self-explanatory, although to be fair to snails, they move in the best way for them.
Vocabulary: register - this word has many meanings as a noun and as a verb; here, the noun, register, means the kind of language that you use in a certain social context, the social tone of the words you are using.
By Mivart, St. George Jackson, 1827-1900 - Dogs, jackals, wolves, and foxes: a monograph of the Canidae, Public Domain, https://commons.wikimedia.org/w/index.php?curid=14904212
