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Proof step 5 from proof modeqp-01 (previous step)

i * j + j0 = i0 * j + j1, abs(j1) < j, abs(j0) < j, j1 ≤ 0, j0 ≤ 0 ⊦ i = i0 ∧ j0 = j1, 0 < j + j1 ∧ 0 < j + j0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs name: Abs-neg seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [j1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ 0 - i = ~ i; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1% ⊦ i + j < i + i0 ↔ j < i0; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i +1 = i + 1; (s)% ⊦ i < i + j ↔ 0 < j; (s)%numlsf-01: ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward) from specification int-abs% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≤ j → (j < i ↔ false); (as)% ⊦ ¬ j < 0 → abs(j) = j; (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)%lf-03: ⊦ i < j ∧ i = i0 → i0 < j; (forward) from specification int-basic1% ⊦ i + 0 = i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s) Used terms: j1