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Proof step 12 from proof modlsp-03 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: ⊦ ¬ i < j → (i = j ↔ ¬ j < i); (ss)% ⊦ i ≤ i + j ↔ 0 ≤ j; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ 0 < j → abs(j) = j; (s)% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ 0 < i ∧ 0 < j → 0 < i + j; (s)% ⊦ j < 0 → abs(j) = 0 - j; (s)% ⊦ 0 - i = ~ i; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i < j → ¬ j < i; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i < ~ j ↔ i + j < 0; (s)% ⊦ i < 0 ∧ j < 0 → i + j < 0; (s)%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-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ i +1 = i + 1; (s)% ⊦ ¬ abs(i) < i; (s) Used terms: i2 + j