Here is the rules for Matthew's Function. The ^ symbol is supposed to represent an up arrow and means the rest of the equation. This is very much related to BEAF, however fails to be as strong as BEAF. A^+^B=A^^^^...^^^A with B arrows A^...^^B=A^...^(A^...^(A^...^(...(A^...^)...))) with B copies of A A^...^^^B=A^...^^(A^...^^(A^...^^(...(A^...^^)...))) with B copies of A A^...^+^B=A^...^^^...^^^A with B arrows A^++^B=A^+^+^...^+^+^A with B +'s A^...^++^B=A^...^+^+^...^+^+^A with B +'s A^+++^B=A^++^++^...^++^++^A with B ++'s A^*^B=A^+++...+++^A with B +'s A^...^*^B=A^...^+++...+++^A with B+'s
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