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Proof step 4 from proof dist-add-01 (previous step)

m0 < m ⊦ dist(m + n, m0 + n) = dist(m, m0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ n < m → dist(m, n) = m - n
Proof step 6:
m0 < m, m0 + n < m + n ⊦ m + n - m0 + n = dist(m, m0)

Rule argument

spec: <toplevel> name: dist-snd seq: ⊦ n < m → dist(m, n) = m - n subst: [n, m] → [m0 + n, m + n] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ ¬ n < n; (s) Used terms: m0 + n, m + n