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Proof step 7 from proof gcdex-dist (previous step)

m * m0 + gcd(m, n) = n * n0 ⊦ gcd(m, n) = dist(m * m0, n * n0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ m ≤ n → dist(m, n) = n - m
Proof step 9:
m * m0 ≤ n * n0, m * m0 + gcd(m, n) = n * n0 ⊦ gcd(m, n) = n * n0 - m * m0

Rule argument

spec: nat-prime inst: name: dist-fst seq: ⊦ m ≤ n → dist(m, n) = n - m subst: [n, m] → [n * n0, m * m0] flags: rewrite, current

Additional information

Used simplifier rules: lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic1%zero-01: ⊦ m0 * m = 0 ↔ m0 = 0 ∨ m = 0; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used terms: n * n0, m * m0