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Proof step 29 from proof int2int-lastn-negpos (previous step)

# bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, bits2int(bits) < 2 ^ n→i(m), bits2int(bits) < 0, m ≤ # int2bits(i), 0 ≤ bits2int(lastn(m, int2bits(i))), 0 < 2 ^ n→i(m), i < 0, i < bits2int(lastn(m, int2bits(i))), i + 1 < 2 ^ n→i(m), i < 2 ^ n→i(m), bits2int(bits) < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < 2 ^ n→i(m) ⊦ bits2int(lastn(m, int2bits(i))) = bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, bits = [], bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ abs(i) < j → i % j = i
Proof step 31:
¬ abs(bits2int(lastn(m, int2bits(i)))) < 2 ^ n→i(m), # bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, bits2int(bits) < 2 ^ n→i(m), bits2int(bits) < 0, m ≤ # int2bits(i), 0 ≤ bits2int(lastn(m, int2bits(i))), 0 < 2 ^ n→i(m), i < 0, i < bits2int(lastn(m, int2bits(i))), i + 1 < 2 ^ n→i(m), i < 2 ^ n→i(m), bits2int(bits) < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < 2 ^ n→i(m) ⊦ bits2int(lastn(m, int2bits(i))) = bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, bits = [], bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) = 0

Rule argument

seq: ⊦ abs(i) < j → i % j = i subst: [j, i] → [2 ^ n→i(m), bits2int(lastn(m, int2bits(i)))] rotate: F spec: int-div inst: name: mod-zero-05

Additional information

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