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

2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, i→n(i) ≤ # int2bits(j / 2 ^ i - 1) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + 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 54:
⊦ bits ≠ [] → # bits.rest = (# bits)-1
Proof step 55:
fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) ≠ [], 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, i→n(i) ≤ # int2bits(j / 2 ^ i - 1) + (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)))-1, j + 2 ^ i ≤ 0, j + 2 ^ i < i ⊦ i→n(i) = (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)))-1, j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i

Rule argument

seq: ⊦ bits ≠ [] → # bits.rest = (# bits)-1 subst: [bits] → [fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i))] rotate: F spec: list-data inst: bitlista name: length-rest

Additional information

Used simplifier rules: ⊦ fillsgn(n, bits) = [] ↔ bits = [] ∧ n = 0;% ⊦ int2bits(j) ≠ []; Used terms: fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i))