[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 54 from proof band-011x-pos (previous step)

bits2int(bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2))) = (j / 2) % 2 ^ i0, j % 2 = 1, 0 < j, 0 ≤ j, 0 ≤ i0, bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2)).first, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ bits2int(bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2)) + true ') = j % (2 ^ i0 * 2), j = 1, ~1 = i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ bits ≠ [] ∧ bits0 ≠ [] ∧ ¬ bits.first ∧ ¬ bits0.first → ¬ bitsand(bits, bits0).first
Proof step 56:
¬(false ' + mklist(true, i→n(i0)) ≠ [] ∧ uint2bits(j / 2) ≠ [] ∧ ¬ (false ' + mklist(true, i→n(i0))).first ∧ ¬ uint2bits(j / 2).first), bits2int(bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2))) = (j / 2) % 2 ^ i0, j % 2 = 1, 0 < j, 0 ≤ j, 0 ≤ i0, bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2)).first, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ bits2int(bitsand(false ' + mklist(true, i→n(i0)), uint2bits(j / 2)) + true ') = j % (2 ^ i0 * 2), j = 1, ~1 = i0

Rule argument

name: and-pos rotate: F subst: [bits0, bits] → [uint2bits(j / 2), false ' + mklist(true, i→n(i0))]

Additional information

Used terms: uint2bits(j / 2), false ' + mklist(true, i→n(i0))