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Proof step 19 from proof uint2bits-mod2pot (previous step)

0 ≤ i0 ∧ 0 ≤ j, 0 ≤ i0, j ≤ 2 ^ (i0 + 1) + 1, 0 < j, 2 ^ i0 * 2 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, j = 1, uint2bits(j % (2 ^ i0 * 2)) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest) + (j % 2 = 1)')

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
0 ≤ i0, 0 ≤ 2 ^ i0, ∀ j0, i. i→n(j0) < i→n(2 ^ i0 * 2 + 1) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, uint2bits((2 ^ i0 * 2 + 1) % (2 ^ i0 * 2)) = false ' + true '

Rule argument

Additional information

Used simplifier rules: uint2bits-2pot: 0 ≤ i ⊦ uint2bits(2 ^ i) = false ' + true ' + mklist(false, i→n(i));% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0);% ⊦ sgn(false ' + true ') = false ' + true ';% ⊦ sgn(false ' + mklist(false, n) + bits) = sgn(false ' + bits);% ⊦ 0 ≤ i → (i * 2 + 1) % 2 = 1;% ⊦ lastn(n, bits + mklist(boolvar, n)) = mklist(boolvar, n);% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ (boolvar ' + bits).rest = bits;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ 0 ≤ i → (2 * i + 1) / 2 = i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ i * 0 = 0;% ⊦ (* i2) * i < (* i1) * i + j0 ↔ (i2 - i1) * i < j0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 ≤ j → 2 ^ j ≠ 0;% ⊦ 2 * i = 0 ↔ i = 0;% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1);% ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;