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Addition is commutative, which easily proves that (freely speaking) the order of summing a list of numbers does not matter; all orders of applying the summation are equal. How would you go about proving, from a notion of minimalist "first principles", that every expression involving only addition and subtraction can have its order of constituents altered arbitrarily and remain equal to its original form?

I'd probably prefer to assume that if an expression starts with some negative value $-1$, we mean to write $0-1$, to avoid having to define the negative integers while making the proof. Or alternatively I'm willing to exclude expressions wherein the first operation happens to be subtraction and not addition.

matanox
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    "every expression involving only addition and substraction can have its order of constituents altered arbitrarily and remain equal to its original form" Not so. $7-11\ne11-7$. – Gerry Myerson Apr 27 '20 at 09:53
  • If you "avoid having to define the negative integers while making the proof", what do you do with (say) $1-2$? If you just say it's not defined, then you have to account for how $1-2+3$ is superficially not defined but $1+3-2$ is. You can say something like "when they are both defined the expressions are equal" but to be honest this sounds more difficult than just introducing negative numbers :) – Ben Millwood Apr 27 '20 at 10:08
  • @GerryMyerson in constituents I meant operation and its argument taken as one unit. I should make that clearer in an edit of the question. – matanox Apr 27 '20 at 10:15
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    In that case, you're just dealing with addition. Some of the addends are positive, some negative, but you're not really doing any subtraction (let alone substraction), you're just doing addition, so commutativity applies. – Gerry Myerson Apr 27 '20 at 10:17
  • @GerryMyerson That's permissible as long as I acknowledge negative integers isn't it. Not sure I would like to however; imagine I would like to prove strictly within the domain of Natural numbers only. In that case we really interpret substraction as substraction, I would posit. Put another way, if we escape infix notation, I mean that each constituent is something like $add(_,y)$ or $substract(_,y)$ where all $y$ are Naturals, and that we need to prove that the order of applying these does not matter, namely that all orderings yield the same. – matanox Apr 27 '20 at 10:26
  • As @Bob writes, you can't really avoid negative numbers. You can't do "seven subtract eight add nine" without doing "seven subtract eight" first. And, please, there is no such word as "substraction", nor "substract". – Gerry Myerson Apr 27 '20 at 10:33

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Since $a-b = a+(-b)$, you can replace each $-x$ by $+(-x)$, reorder the terms as you like, then replace each $+(-x)$ by $-x$.

For example

$4 + 3 - 2 - 1 = 4+3+(-2)+(-1)=3+(-1)+4+(-2)=3-1+4-2$

gandalf61
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