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Proof step 45 from proof gcd-rec-sub-03 (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 46:
abs(i) - (abs(i) - abs(i0)) = abs(i0), 0 < abs(i) - abs(i0) ∧ abs(i) - abs(i0) < abs(i), i < 0 ⊦ abs(i) < i0, i0 < 0, gcd(abs(i), abs(i) - (abs(i) - abs(i0))) = gcd(abs(i) - abs(i0), abs(i0)), 0 < i ∧ i < i0 + i

Rule argument

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

Additional information

Used simplifier rules: abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%Abs-pos: ⊦ ¬ j < 0 → abs(j) = j; from specification int-abs%subsub: ⊦ ~ i - j = ~(i + j); from specification int-abs%sub-minus: ⊦ ~ i - ~ j = j - i; from specification int-abs%sub-add: ⊦ i + j - i = j; 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 Used formulas: abs(i) - (abs(i) - abs(i0)) = abs(i0)