[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 71 from proof cast-mod-neg-high (previous step)

i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 72:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 73:
¬ i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i
Proof step 94:
i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), i→n(i) = # int2bits(j % 2 ^ i).rest, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(int2bits(j % 2 ^ i).rest) = j % 2 ^ i

Rule argument

seq: ⊦ n ≤ # bits → fillsgn(n, bits) = bits subst: [bits, n] → [int2bits(j % 2 ^ i), i→n(i) + 1] rotate: F spec: bitopsaux inst: name: fillsgn-longer

Additional information

Used terms: int2bits(j % 2 ^ i), i→n(i) + 1