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Proof step 7 from proof mod-pred-02 (previous step)

j0 < j, 0 < j, 0 ≤ j0, 0 ≤ i1 ⊦ ((i1 * j + j0) + 1) % j = 0, ((i1 * j + j0) + 1) % j = j0 + 1

User applied rule insert elim lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j)
Proof step 9:
(i1 * j + j0) + 1 = i0 * j + j1 ∧ 0 ≤ j1 ∧ j1 < j ∧ 0 ≤ (i1 * j + j0) + 1, j0 < j, 0 < j, 0 ≤ j0, 0 ≤ i1 ⊦ j1 = 0, j1 = j0 + 1

Rule argument

spec: int-div inst: name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [(i1 * j + j0) + 1, j, i, j1] flags: current, []

Additional information

Used simplifier rules: numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%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%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (i1 * j + j0) + 1, j, i, j1