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Proof step 30 from proof uint2bits-mod2pot (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ 0 < i ∧ i ≠ 1 → uint2bits(i) = uint2bits(i / 2) + (i % 2 = 1)
Proof step 32:
¬(0 < (j / 2) % 2 ^ i0 * 2 + 1 ∧ (j / 2) % 2 ^ i0 * 2 + 1 ≠ 1), uint2bits((j / 2) % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest)), j % 2 = 1, 2 ^ i0 * 2 < j, 0 < j, 0 ≤ i0, 0 < 2 ^ i0, 0 ≤ 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)) ⊦ uint2bits((j / 2) % 2 ^ i0 * 2 + 1) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest) + true '), j = 1, ~1 = i0
Proof step 47:
0 < (j / 2) % 2 ^ i0 * 2 + 1 ∧ (j / 2) % 2 ^ i0 * 2 + 1 ≠ 1, uint2bits((j / 2) % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest)), j % 2 = 1, 2 ^ i0 * 2 < j, 0 < j, 0 ≤ i0, 0 < 2 ^ i0, 0 ≤ 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)) ⊦ uint2bits(((j / 2) % 2 ^ i0 * 2 + 1) / 2) + (((j / 2) % 2 ^ i0 * 2 + 1) % 2 = 1) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest) + true '), j = 1, ~1 = i0

Rule argument

name: uint2bits-rec-01 rotate: F subst: [i] → [(j / 2) % 2 ^ i0 * 2 + 1]

Additional information

Used terms: (j / 2) % 2 ^ i0 * 2 + 1