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Proof step 11 from proof gcdex-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
gcd(i, j) = i * i0 + j * j0, gcd(i, j) = i * i0 * (j + 1) + j * (j0 - i0 * i), j < 0, i0 < 0 ⊦ j = 0, ∃ i0, j0. gcd(i, j) = i * i0 + j * j0 ∧ 0 ≤ i0

Rule argument

Additional information

Used simplifier rules: neg-15: ⊦ i + ~ j = i - j; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%succ-rmerge: ⊦ i * j + i = i * (j + 1); from specification int-mult%subneg: ⊦ i - ~ j = i + j; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs