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Proof step 19 from proof gcdex-pos (previous step)

gcd(i, j) = i * i0 + j * j0, 0 < j, i0 < 0, gcd(i, j) = i * (i0 - i0 * j) + j * (j0 + i0 * i), ∀ i1. gcd(i, j) = i * (i0 - i1 * j) + j * (j0 + i1 * i) ⊦ j = 0, ∃ i0, j0. gcd(i, j) = i * i0 + j * j0 ∧ 0 ≤ i0

User applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: i0 - i0 * j, j0 + i0 * i abort: F discard: T computed: T

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%le-same-02: ⊦ i < 0 → (i * j ≤ i ↔ 1 ≤ j); from specification int-mult%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i0 - i0 * j, j0 + i0 * i