Rewrite rules for Booleans

Boolean expressions b can be composed of

true false // Constants
!b // Negation
b && b // Conjunction
b || b // Disjunction

and of the usual comparison operations:

e <= e, e >= e, e < e, e > e, e == e, e != e

Here are reduction rules for Boolean expressions (e is an arbitrary expression):

!true --> false
!false --> true
true && e --> e
false && e --> false
true || e --> true
false || e --> e

Note that && and || do not always need their right operand to be evaluated.
We say, these expressions use “short-circuit evaluation”.

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