Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := 0\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y\) equal to \(1\) if \(x=y=0\) and \(2\) otherwise.

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y\) equal to \(3\) if \(x=1\) and \(y=2\) and \(4\) otherwise.

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x+1\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x+y\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x \cdot y + 1\).

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y\) equal to \(1\) if \(x=0\) and \(y \leq 2\), \(2\) if \(x=0\) and \(y{\gt}2\), and \(x\) otherwise.

Use the natural numbers \(\mathbb {N}\) with operation \(x \circ y := x + y\).

Use the boolean type \(\mathrm{Bool}\) with \(x \circ y := x || y\).

Use the natural numbers \(\mathbb {N}\) with \(x \circ y := x+y\).

Use the reals \(\mathbb {R}\) with \(x \circ y := (x+y)/2\).

Use the natural numbers \(\mathbb {N}\) with \(x \circ y := y\).

Use the natural numbers \(\mathbb {N}\) with \(x \circ y\) equal to \(x\) when \(x=y\) and \(x+1\) otherwise.

Use the natural numbers \(\mathbb {N}\) with \(x \circ y := 0\).

Use the natural numbers \(\mathbb {N}\) with \(x \circ y := x+y\).