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Proof step 60 from proof div-div-rec (previous step)

2 ^ i * j1 + j0 < i, 2 ^ i * j1 + j0 ≤ 0, 2 ^ i * 2 ≤ abs(2 ^ i * j1 + j0), 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 < i, 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 ≤ 0, 0 ≤ i, (2 * i1 + j1) + 1 < i, 2 ^ i * ((2 * i1 + j1) + 1) + j0 < 0, 2 * i1 + j1 < 0, j0 < i, j0 ≤ 0, abs(j1) < 2, i1 < i, i1 ≤ 0, j1 < i, j1 ≤ 0, abs(j0) < 2 ^ i, 0 < i ⊦ i1 = (2 ^ i * (j1 + 2 * i1) + j0) / (2 ^ i * 2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 61:
⊦ i ≠ 1 ∧ 0 ≤ i → (i < 2 ↔ i = 0)
Proof step 62:
¬(abs(j1) ≠ 1 ∧ 0 ≤ abs(j1)), 2 ^ i * j1 + j0 < i, 2 ^ i * j1 + j0 ≤ 0, 2 ^ i * 2 ≤ abs(2 ^ i * j1 + j0), 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 < i, 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 ≤ 0, 0 ≤ i, (2 * i1 + j1) + 1 < i, 2 ^ i * ((2 * i1 + j1) + 1) + j0 < 0, 2 * i1 + j1 < 0, j0 < i, j0 ≤ 0, abs(j1) < 2, i1 < i, i1 ≤ 0, j1 < i, j1 ≤ 0, abs(j0) < 2 ^ i, 0 < i ⊦ i1 = (2 ^ i * (j1 + 2 * i1) + j0) / (2 ^ i * 2)

Rule argument

spec: int-abs inst: name: lstwo seq: ⊦ i ≠ 1 ∧ 0 ≤ i → (i < 2 ↔ i = 0) subst: [i] → [abs(j1)] flags: rewrite

Additional information

Used simplifier rules: ls-mult-2pot: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 → (j * 2 ^ i < 2 ^ i0 ↔ j < 2 ^ (i0 - i));%abs-07: ⊦ abs(i) = 0 ↔ i = 0; from specification int-abs%abs-03: ⊦ 0 ≤ abs(i); from specification int-abs%add-zero: ⊦ i + 0 = i; 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%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; 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%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%sub-sub-03: ⊦ i - i = 0; from specification int-abs Used terms: abs(j1)