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Proof step 43 from proof gcd-rec-sub-01 (previous step)

0 < abs(i0) ∧ abs(i0) < abs(i), abs(i0) < abs(i), i < 0, 0 < i0 ⊦ gcd(abs(i), abs(i0 - abs(i))) = gcd(abs(i0), abs(i) - abs(i0)), i = 0

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: abs(i0 - abs(i)) = abs(i) - abs(i0)

Additional information

Used simplifier rules: gcd-neg: ⊦ gcd(~ i, j) = gcd(i, j);%gcd-com: ⊦ gcd(i, j) = gcd(j, i);%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%subneg: ⊦ i - ~ j = i + j; from specification int-abs%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%subsub: ⊦ ~ i - j = ~(i + j); from specification int-abs%negeq: ⊦ ~ i = j ↔ i + j = 0; from specification int-abs%abseq: ⊦ abs(i) + i = 0 ↔ i ≤ 0; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used formulas: abs(i0 - abs(i)) = abs(i) - abs(i0)