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Proof step 21 from proof int2int-lastn-negpos (previous step)

bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, int2bits(i) = bits + lastn(m, int2bits(i)), # bits = # int2bits(i) - m, m ≤ # int2bits(i), i < 0, bits.first ⊦ bits = [], m = 0, i % 2 ^ n→i(m) = 0, bits2int(lastn(m, int2bits(i))) = (bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i)))) % 2 ^ n→i(m) + 2 ^ n→i(m), lastn(m, int2bits(i)).first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
i0 * j + j0 < 0, 0 < j, 0 ≤ j0, j0 % j ≠ 0 ⊦ (i0 * j + j0) % j = j0 % j - j
Proof step 23:
¬ ( bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) < 0 ∧ 0 < 2 ^ n→i(m) ∧ 0 ≤ bits2int(lastn(m, int2bits(i))) ∧ bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) ≠ 0), bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, int2bits(i) = bits + lastn(m, int2bits(i)), # bits = # int2bits(i) - m, m ≤ # int2bits(i), i < 0, bits.first ⊦ bits = [], m = 0, i % 2 ^ n→i(m) = 0, bits2int(lastn(m, int2bits(i))) = (bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i)))) % 2 ^ n→i(m) + 2 ^ n→i(m), lastn(m, int2bits(i)).first
Proof step 28:
bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) < 0 ∧ 0 < 2 ^ n→i(m) ∧ 0 ≤ bits2int(lastn(m, int2bits(i))) ∧ bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) ≠ 0, bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, int2bits(i) = bits + lastn(m, int2bits(i)), # bits = # int2bits(i) - m, m ≤ # int2bits(i), i < 0, bits.first ⊦ bits = [], m = 0, i % 2 ^ n→i(m) = 0, bits2int(lastn(m, int2bits(i))) = (bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) - 2 ^ n→i(m)) + 2 ^ n→i(m), lastn(m, int2bits(i)).first

Rule argument

seq: i0 * j + j0 < 0, 0 < j, 0 ≤ j0, j0 % j ≠ 0 ⊦ (i0 * j + j0) % j = j0 % j - j subst: [j0, j, i0] → [bits2int(lastn(m, int2bits(i))), 2 ^ n→i(m), bits2int(bits)] rotate: F spec: int-div inst: name: modeq-neg-02

Additional information

Used terms: bits2int(lastn(m, int2bits(i))), 2 ^ n→i(m), bits2int(bits)