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Proof step 14 from proof square-interval (previous step)

2 * (i * (i + 1)) / 2 ≤ 2 * ((j * (j + 1)) / 2 + j0), (j * (j + 1)) % 2 ≠ 1 ∧ 0 < j * (j + 1), ((i + 1) * (i + 2)) % 2 ≠ 1 ∧ 0 < (i + 1) * (i + 2), j * (j + 1) + 2 * j0 < (i + 1) * (i + 2), 0 ≤ j0, 0 ≤ j, 0 ≤ i, j0 ≤ j, (j * (j + 1)) / 2 + j0 < ((i + 1) * (i + 2)) / 2, (i * (i + 1)) / 2 ≤ (j * (j + 1)) / 2 + j0 ⊦ i * (i + 1) ≤ j * (j + 1) + 2 * j0, i = j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div2zero seq: ⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i subst: [i] → [i * (i + 1)] flags: rewrite, current

Additional information

Used simplifier rules: square-even-02: ⊦ ((i + 1) * (i + 2)) % 2 = 0;%square-even: ⊦ 0 ≤ i → (i * (i + 1)) % 2 = 0;%div-lezero: ⊦ 0 < j → (i / j ≤ 0 ↔ i < j);%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%le-pos-01: ⊦ 0 < i → (j * i ≤ i0 * i ↔ j ≤ i0); from specification int-mult%zero-gr-02: ⊦ 0 ≤ i → (0 < i * j ↔ ¬ ¬(0 < j ∧ i ≠ 0)); from specification int-mult%numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; 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%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%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-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%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%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%ls-dist-03: ⊦ i * (j + 1) + j1 < i ↔ i * j + j1 < 0; from specification int-mult%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%numls-03: ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%lep-01: ⊦ i + j ≤ i ↔ j ≤ 0; from specification int-abs%square-ls-04: ⊦ 0 < i → ¬ i * (i + 1) < 2; from specification int-mult%lst-01: ⊦ ¬ i < j → (i ≠ j ↔ j < i); from specification int-basic1%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i * (i + 1)