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Proof step 6 from proof mod-succ-01 (previous step)

0 < j, j0 < j, 0 ≤ j0, 0 ≤ i0 ⊦ ((i0 * j + j0) + 1) % j = j0 + 1, j0 + 1 = j

User applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [(i0 * j + j0) + 1, j, i, j1] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; (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 ↔ ¬ j < i; (ss)%numlsf: ⊦ i < (* j) ∧ j < 0 → i < 0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ i + j < j ↔ i < 0; (s)% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)% ⊦ i ≤ j → ¬ j < i; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (i0 * j + j0) + 1, j, i, j1