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Proof step 11 from proof modlep-03 (previous step)

(i2 + j1) + 1 < abs(i2), (i2 + j1) + 1 < abs(i2 + j), abs(j1) < abs(i2), i2 + j1 < abs(i2), i2 + j1 < abs(i2 + j), j ≤ abs(i2), j * (i + 1) + i2 ≤ j * i0, (i2 + j1) + 1 < j, (i2 + j) + j1 < abs(i2 + j), j * i + i2 + j ≤ j * i0, 0 < j, j ≤ abs(i2 + j), i0 < i, abs(j1) < j, 0 ≤ (i2 + j) + j1, abs(j1) < abs(i2 + j), i2 + j1 < 0, i2 + j1 < j ⊦ i2 + j = 0, i2 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-pos seq: ⊦ ¬ j < 0 → abs(j) = j subst: [j] → [i2 + j] flags: rewrite

Additional information

Used simplifier rules: lst-01: ⊦ ¬ i < j → (i ≠ j ↔ j < i); from specification int-basic1%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%lep: ⊦ i ≤ i + j ↔ 0 ≤ j; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%zero-02: ⊦ 0 < i ∧ 0 < j → 0 < i + j; from specification int-basic2%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs%zero-01: ⊦ i < 0 ∧ j < 0 → i + j < 0; from specification int-basic2%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; 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%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%numlsf: ⊦ i < (* j) ∧ j < 0 → i < 0; (forward) from specification int-basic1%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%lst-04: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0; from specification int-basic2%absls-01: ⊦ ¬ abs(i) < i; from specification int-abs Used terms: i2 + j