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Proof step 17 from proof cast-mod-pos-low (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
j % 2 ^ i < 2 ^ (i - 1), 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, j % 2 ^ i < j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ i ≤ j, i→n(i) ≤ # int2bits(j / 2 ^ i) + # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i

Rule argument

Additional information

Used simplifier rules: ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;