[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 29 from proof div-two-intro (previous step)

0 ≤ i0 ⊦ (i0 * 2 + 1) / (j0 * 2) = i0 / j0, j0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 31:
0 ≤ i0 ⊦ (i0 * 2 + 1) / (j0 * 2) = i0 / j0, j0 = 0, (i0 * 2) % (j0 * 2) = 0 ∧ 0 ≤ i0 * 2 ∧ 0 ≤ 1 ∧ 0 < j0 * 2

Rule argument

spec: <toplevel> name: div-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [j0 * 2, 1, i0 * 2] flags: <all paths>

Additional information

Used simplifier rules: mod-mult-02: 0 ≠ j, i0 ≠ 0 ⊦ (i0 * i) % (i0 * j) = i % j * i0;%div-mult-01: j ≠ 0, i0 ≠ 0 ⊦ (i * i0) / (j * i0) = i / j;%div-eq-zero: ⊦ 0 < i ∧ 0 < j → (i / j = 0 ↔ i < j);%nummult-ls-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ 2 * i = 0 ↔ i = 0; (s)% ⊦ i = i + j ↔ j = 0; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≠ j → (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% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i < j → ¬ j < i; (s) Used terms: j0 * 2, 1, i0 * 2