[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 11 from proof square-scale (previous step)

i % 10 = 0, ∀ i1, i0. i→n(i1) < i→n(i) → true → even(mkDecimal(i1 * i1, 2 * abs(i0)).scale) ⊦ i = 0, i0 = 0, even(mkDecimal(i * i, 2 * abs(i0)).scale)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ i % 10 = 0 ∧ j ≠ 0 ∧ i ≠ 0 → mkDecimal(i, j).scale = mkDecimal(i / 10, abs(j) - 1).scale
Proof step 13:
(i * i) % 10 = 0 ∧ 2 * abs(i0) ≠ 0 ∧ i * i ≠ 0, i % 10 = 0, ∀ i1, i0. i→n(i1) < i→n(i) → true → even(mkDecimal(i1 * i1, 2 * abs(i0)).scale) ⊦ i = 0, i0 = 0, even(mkDecimal((i * i) / 10, abs(2 * abs(i0)) - 1).scale)

Rule argument

spec: decimal inst: name: scale-rec seq: ⊦ i % 10 = 0 ∧ j ≠ 0 ∧ i ≠ 0 → mkDecimal(i, j).scale = mkDecimal(i / 10, abs(j) - 1).scale subst: [j, i] → [2 * abs(i0), i * i] flags: rewrite

Additional information

Used simplifier rules: mod-square-10: ⊦ (i * i) % 10 = 0 ↔ i % 10 = 0; from specification int-gcd%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%two-zero: ⊦ 2 * i = 0 ↔ i = 0; from specification int-mult%abs-07: ⊦ abs(i) = 0 ↔ i = 0; from specification int-abs%square-zero: ⊦ i * i = 0 ↔ i = 0; from specification int-mult Used terms: 2 * abs(i0), i * i