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Proof step 16 from proof mod-add-zero (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic2 inst: name: zero-03 seq: ⊦ 0 < i ∧ ¬ j < 0 → 0 < i + j subst: [j, i] → [j * i1, j0] flags: rewrite

Additional information

Used simplifier rules: zero-neq-01: ⊦ 0 < i → (i * j = 0 ↔ j = 0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%gr-def: ⊦ i > j ↔ j < i; from specification int-abs%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j * i1, j0