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

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 51:
⊦ n = # bits → lastn(n, bits0 + bits) = bits
Proof step 52:
i→n(i) ≠ # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j + 2 ^ i < i, j + 2 ^ i ≤ 0, i→n(i) ≤ # int2bits(j / 2 ^ i - 1) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j < j % 2 ^ i, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < 0, 0 < i, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0
Proof step 70:
i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j + 2 ^ i < i, j + 2 ^ i ≤ 0, i→n(i) ≤ # int2bits(j / 2 ^ i - 1) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j < j % 2 ^ i, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < 0, 0 < i, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ bits2int(fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest) = j % 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0

Rule argument

seq: ⊦ n = # bits → lastn(n, bits0 + bits) = bits subst: [bits, bits0, n] → [fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, int2bits(j / 2 ^ i - 1), i→n(i)] rotate: F spec: list-del inst: bitlista name: lastN-append

Additional information

Used terms: fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, int2bits(j / 2 ^ i - 1), i→n(i)