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Proof step 32 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), 2 ^ n→i(m) ≤ bits2int(lastn(m, int2bits(i))), 0 < bits2int(lastn(m, int2bits(i))), i + 1 < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < bits2int(lastn(m, int2bits(i))) ⊦ 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 33:
⊦ 0 < i ∧ 0 ≤ j → (2 ^ j ≤ i ↔ ¬ # int2bits(i) ≤ i→n(j) + 1)
Proof step 34:
0 < bits2int(lastn(m, int2bits(i))) ∧ 0 ≤ 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), ¬ # int2bits(bits2int(lastn(m, int2bits(i)))) ≤ i→n(n→i(m)) + 1, 0 < bits2int(lastn(m, int2bits(i))), i + 1 < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < bits2int(lastn(m, int2bits(i))) ⊦ 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

name: int2bits-lelen-01 rotate: F subst: [i, j] → [bits2int(lastn(m, int2bits(i))), n→i(m)]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ 0 ≤ n→i(n);% ⊦ i < (* j) ∧ j < 0 → i < 0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i * j = j * i;% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ 0 ≤ i ∧ i ≤ 1 → i = 0 ∨ i = 1; (forward) Used terms: bits2int(lastn(m, int2bits(i))), n→i(m)