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Proof step 25 from proof mod-ls-same-01 (previous step)

0 < i, j < 0 ⊦ i % j < i ↔ ¬ i + j < 0, ~1 = j, i = 0, j = 1

User applied rule insert elim lemma

And obtained the following new goals

Proof step 26:
⊦ j < 0 ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) ∧ i0 ≤ 0)
Proof step 27:
i = i0 * j + j0, j < 0, 0 < i, j0 < abs(j), 0 ≤ j0, i0 ≤ 0, 0 ≤ i ⊦ j0 < i ↔ ¬ i + j < 0, ~1 = j, i = 0, j = 1

Rule argument

spec: <toplevel> name: div-elim-posneg seq: ⊦ j < 0 ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i, j, i0, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s) Used terms: i, j, i0, j0