you made a very major mistake...
uhh
\( f^{\diamond - n}(f^{\diamond n + 1}(x)) \neq f^{\diamond 1}(x) \)
this is exactly the assumption I was describing doesn't work.
\( (f^{\diamond n} \circ f^{\diamond m})(x) \neq f^{\diamond m +n}(x) \)
instead, we'd need to create a new operator
\( (f^{\diamond n} \diamond f^{\diamond m})(x) = f^{\diamond m + n}(x) \)
the diamond operator is NOT composition. And the diamond superscript is an index of superfunctions.
\( f^{\diamond n}(x) \neq (f_1 \circ f_2 \circ f_3 ... f_n)(x) = f^{\circ n}(x) \)
instead, it's defined by
\( f(x) = f^{\diamond 1}(g^{\diamond 1}(x) + 1) \)
\( f^{\diamond 1}(x) = f^{\diamond 2}(g^{\diamond 2}(x) + 1) \)
continued arbitrarily
where
\( g^{\diamond n}(f^{\diamond n}(x)) = x \)
seriously. You could try reading first.
my definitions are perfectly consistent. I'll give you the benefit of the doubt for maybe misreading \( \diamond \) for \( \circ \)
then again maybe you should try actually reading instead of eating chocolate... and thinking instead of claiming other people arent...
uhh
\( f^{\diamond - n}(f^{\diamond n + 1}(x)) \neq f^{\diamond 1}(x) \)
this is exactly the assumption I was describing doesn't work.
\( (f^{\diamond n} \circ f^{\diamond m})(x) \neq f^{\diamond m +n}(x) \)
instead, we'd need to create a new operator
\( (f^{\diamond n} \diamond f^{\diamond m})(x) = f^{\diamond m + n}(x) \)
the diamond operator is NOT composition. And the diamond superscript is an index of superfunctions.
\( f^{\diamond n}(x) \neq (f_1 \circ f_2 \circ f_3 ... f_n)(x) = f^{\circ n}(x) \)
instead, it's defined by
\( f(x) = f^{\diamond 1}(g^{\diamond 1}(x) + 1) \)
\( f^{\diamond 1}(x) = f^{\diamond 2}(g^{\diamond 2}(x) + 1) \)
continued arbitrarily
where
\( g^{\diamond n}(f^{\diamond n}(x)) = x \)
seriously. You could try reading first.
my definitions are perfectly consistent. I'll give you the benefit of the doubt for maybe misreading \( \diamond \) for \( \circ \)
then again maybe you should try actually reading instead of eating chocolate... and thinking instead of claiming other people arent...