
Re: Two Finite Arithmetics
Posted:
Apr 8, 2014 11:03 PM


On Tue, 8 Apr 2014, Dan Christensen wrote:
> Here, I have translated your axioms into DC Proof format. Download my software, enter these axioms (the "@" is mapped to epsilon) and see what you can do with them. You have to construct your own proofs one line at a time, but the software will not let you make mistakes. Instant feedback is provided with each line you enter it. You might start by proving the associativity and commutativity of + and * as you have defined them. You can get started using with the tutorial or user manual.
I will not.
> 1 Set(n) > Axiom > > 2 0 @ n > Axiom > > 3 m @ n > Axiom > > 4 ALL(a):[a @ n => s(a) @ n] > Axiom > > 5 s(m)=m > Axiom > > 6 ALL(a):~s(a)=0 > Axiom > > 7 ALL(a):ALL(b):[a @ n & b @ n & ~a=m & ~b=m => [s(x)=s(y) => x=y]] > Axiom > > 8 ALL(a):[Set(a) & ALL(b):[b @ a => b @ n] > => [0 @ a & ALL(b):[b @ a => s(b) @ a] > => ALL(b):[b @ n => b @ a]]] > Axiom > > 9 ALL(a):ALL(b):[a @ n & b @ n => a+b @ n] > Axiom > > 10 ALL(a):[a @ n => 0+a=a] > Axiom > > 11 ALL(a):ALL(b):[a @ n & b @ n => s(a)+b=s(a+b)] > Axiom > > 12 ALL(a):ALL(b):[a @ n & b @ n => a*b @ n] > Axiom > > 13 ALL(a):[a @ n => 0*a=0] > Axiom > > 14 ALL(a):[a @ n & b @ n => s(a)*b=a*b+b] > Axiom > >

