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Proof step 11 from proof mod-pred-neg-01 (previous step)

i0 * j + j0 = (i2 * j + j1) + 1, j0 < j, i0 < j, 0 < j, abs(j0) < j, i2 * j + j1 < ~1, i2 * j + j1 < 0, i2 < j, i2 < j + j1, j1 < 0, j0 < j + j1, i0 < j + j1, 0 < j + j1, j0 ≤ 0, i0 ≤ 0, i2 ≤ 0 ⊦ j0 = j1 + 1, j1 = 0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult name: modeqp seq: i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1 subst: [i, j, j0, i0, j1] → [i0, j, ~(j1 + 1), i2, ~ j0]

Additional information

Used simplifier rules: ⊦ ~ i = ~ j ↔ i = j; (s)% ⊦ 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% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + (j - i0) = i + j - i0; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; (s)% ⊦ i ≤ ~1 ↔ i < 0; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ 0 < i ∧ 0 < j → 0 < i + j; (s)% ⊦ i + 0 = i; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)%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% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≤ j → ¬ j < i; (s) Used terms: i0, j, ~(j1 + 1), i2, ~ j0