[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 3 from proof div-elim3 (previous step)

i < 0, j < 0 ⊦ i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ j < j0

User applied rule insert lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 < i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j)

Rule argument

spec: <toplevel> name: div-elim seq: ⊦ 0 < j ∧ 0 < i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [j, i, i0, j0] → [~ j, ~ i, i0, ~ j0]

Additional information

Used simplifier rules: neg-01: j ≠ 0 ⊦ ~ i / j = ~(i / j);%neg: j ≠ 0 ⊦ i / ~ j = ~(i / j);%mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%mod-neg: j ≠ 0 ⊦ i % ~ j = i % j;% ⊦ 0 < ~ i ↔ i < 0; (s)% ⊦ ~ i = 0 ↔ i = 0; (s)% ⊦ ~ i = j ↔ i + j = 0; (s)% ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (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; (s)% ⊦ i < j → (j ≤ i ↔ false); (as)% ⊦ ~ ~ i = i; (s)% ⊦ ~ i = ~ j ↔ i = j; (s)% ⊦ ~ i * j = ~(i * j); (s)% ⊦ ~ i + ~ j = ~(i + j); (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ ~ j < ~ i ↔ i < j; (s) Used terms: ~ j, ~ i, i0, ~ j0