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Editorial Team
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Editorial Team
Asked: June 13, 20262026-06-13T18:47:23+00:00 2026-06-13T18:47:23+00:00

How do you prove: forall m n : Z, m < n -> m

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How do you prove: forall m n : Z, m < n -> m -n < O in Coq? Many Thanks!

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  1. Editorial Team
    Editorial Team
    2026-06-13T18:47:24+00:00Added an answer on June 13, 2026 at 6:47 pm

    If you just care about proving it, and not about the proof, just use omega:

    Require Import Omega.
    
    Goal forall m n : Z, (m < n)%Z -> (m - n < 0%Z)%Z.
    intros. omega.
    Qed.
    

    If you have to prove this as part of exercises, or a homework, it is not too hard if you rely on some existing proofs.

    For instance, you can combine these guys:

    Zminus_diag_reverse
         : forall n : Z, 0%Z = (n - n)%Z
    
    Zplus_lt_le_compat
         : forall n m p q : Z, (n < m)%Z -> (p <= q)%Z -> (n + p < m + q)%Z
    

    There are definitely more than one way to do it, and it is not a very hard goal if you use some existing lemmas.

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