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Proof step 2 from proof square-zero-02 (previous step)

(i + j) * (i + j + 1) < 0, 0 ≤ j, 0 ≤ i ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: square-zero-01 seq: ⊦ 0 ≤ i → ¬ (i + 1) * (i + 2) ≤ 0 subst: [i] → [i + j - 1]

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s)% ⊦ 1 ≤ i ↔ 0 < i; (s)% ⊦ i + (j - i) = j; (s)% ⊦ i + (j - i0) = i + j - i0; (s)% ⊦ (* i) + i0 - (* j) = (i - j) + i0; (s)%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ i + j ≤ i ↔ j ≤ 0; (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s) Used terms: i + j - 1