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properties of Ackermann function

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The proof 4 A(x,y)<A(x-1,A(x,y))Anormal-⁢xyAnormal-⁢x1Anormal-⁢xyA⁢(x,y)<A⁢(x-1,A⁢(x,y)) use as assumptionPlanetmathPlanetmath the property 5

The proof of property 5 on A(x+1,y)<A(x,A(x+1,))fragmentsAfragmentsnormal-(x1normal-,ynormal-)Afragmentsnormal-(xnormal-,Afragmentsnormal-(x1normal-,normal-)normal-)A⁢(x+1,y)<A⁢(x,A⁢(x+1,)) uses as assumption the property 7.

The proof of property 7 is wrong in the inductive step if y>0y0y>0 you should prove that A(x+1,y)<A(x+2,y)Ax1yAx2yA(x+1,y)<A(x+2,y) instead it prove that A(x+1,y)<A(x+1,y+1)Ax1yAx1y1A(x+1,y)<A(x+1,y+1).

here a valid proof http://logic.amu.edu.pl/images/c/cd/Ackermanntaylor.pdf

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