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Proof step 3 from proof modeqp-neg (previous step)

i * i0 + j0 = i1 * j + j1, i < 0, j0 < 0, j1 < 0, i1 < i, 0 < j + j0, 0 < j + j1, i0 ≤ j, 0 ≤ i0, 0 ≤ j ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: modneqp seq: i * i0 + j0 = i1 * j + j1, 0 ≤ j1, j1 < j, 0 ≤ j0, j0 < j, 0 ≤ i0, i0 ≤ j, 0 ≤ i, i < i1 ⊦ subst: [i, i0, j0, i1, j, j1] → [~ i, i0, ~ j0, ~ i1, j, ~ j1]

Additional information

Used simplifier rules: minus-left: ⊦ ~ i * j = ~(i * j);%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;% ⊦ ~ i + ~ j = ~(i + j); (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ ~ i = ~ j ↔ i = j; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + 0 = i; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ ~ j < ~ i ↔ i < j; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s) Used terms: ~ i, i0, ~ j0, ~ i1, j, ~ j1