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Proof step 32 from proof exdivmod (previous step)

i + i0 * j + j0 = 0, j0 < j, i < 0, 0 < j, 0 ≤ j0, 0 ≤ i0, i ≤ 0 ⊦ i0 * j + i = 0, j - j0 < j

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

spec: int-abs name: elim-sub seq: ⊦ i0 = i - j ↔ i = i0 + j subst: [i, j, i0] → [j, j0, i1]

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%zero-left: ⊦ 0 * i = 0;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i + j ≤ i ↔ j ≤ 0; (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i + 0 = i; (s)% ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: j, j0, i1