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Proof step 8 from proof noeth-modlep (previous step)

j = ~1 - j0, j0 + j = ~1, j * i0 + i0 ≤ i + j * i + j0, 0 ≤ j + j2, j < 0, j * i0 ≤ j * i + j0, i0 < i, 0 ≤ j0, 0 < j2, j0 < j2, ∀ j, i, i0, j0. abs(j) < j2 → abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) → i ≤ i0 ⊦ j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
(~1 - j0) * i0 ≤ (~1 - j0) * i + j0, i0 < i, 0 ≤ j0, 0 < j2, (~1 - j0) * i0 + i0 ≤ i + (~1 - j0) * i + j0, j0 < j2, ∀ j, i, i0, j0. abs(j) < j2 → abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) → i ≤ i0 ⊦ ~1 = j0

Rule argument

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%minus-add: i < 0 ⊦ (* i) + j = j - ~ i; from specification int-abs%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs