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Proof step 30 from proof band-min (previous step)

# int2bits(i2 / 2) + 1 = # int2bits(j), i0 + 2 ^ n→i(# int2bits(j)) ≤ 0, j ≤ 0, bits2int(bitsand(int2bits(i2 / 2) + (~1 = (i2 + 1) % 2)', int2bits(j))) < i0, i2 < ~1, i2 < 0, ∀ i0, j, i1. i→n(i1) < i→n(i2 + 1) → # int2bits(i1) = # int2bits(j) ∧ bits2int(bitsand(int2bits(i1), int2bits(j))) < i0 ∧ i1 ≤ 0 ∧ j ≤ 0 ∧ i0 + 2 ^ n→i(# int2bits(j)) ≤ 0 → false ⊦ ~2 = i2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ j < 0 ∧ j ≠ ~1 → int2bits(j) = int2bits((j - 1) / 2) + (~1 = j % 2)'
Proof step 32:
¬(j < 0 ∧ j ≠ ~1), # int2bits(i2 / 2) + 1 = # int2bits(j), i0 + 2 ^ n→i(# int2bits(j)) ≤ 0, j ≤ 0, bits2int(bitsand(int2bits(i2 / 2) + (~1 = (i2 + 1) % 2)', int2bits(j))) < i0, i2 < ~1, i2 < 0, ∀ i0, j, i1. i→n(i1) < i→n(i2 + 1) → # int2bits(i1) = # int2bits(j) ∧ bits2int(bitsand(int2bits(i1), int2bits(j))) < i0 ∧ i1 ≤ 0 ∧ j ≤ 0 ∧ i0 + 2 ^ n→i(# int2bits(j)) ≤ 0 → false ⊦ ~2 = i2
Proof step 35:
j < 0 ∧ j ≠ ~1, # int2bits(i2 / 2) + 1 = #(int2bits((j - 1) / 2) + (~1 = j % 2)'), i0 + 2 ^ n→i(#(int2bits((j - 1) / 2) + (~1 = j % 2)')) ≤ 0, j ≤ 0, bits2int(bitsand(int2bits(i2 / 2) + (~1 = (i2 + 1) % 2)', int2bits((j - 1) / 2) + (~1 = j % 2)')) < i0, i2 < ~1, i2 < 0, ∀ i0, j, i1. i→n(i1) < i→n(i2 + 1) → # int2bits(i1) = # int2bits(j) ∧ bits2int(bitsand(int2bits(i1), int2bits(j))) < i0 ∧ i1 ≤ 0 ∧ j ≤ 0 ∧ i0 + 2 ^ n→i(# int2bits(j)) ≤ 0 → false ⊦ ~2 = i2

Rule argument

seq: ⊦ j < 0 ∧ j ≠ ~1 → int2bits(j) = int2bits((j - 1) / 2) + (~1 = j % 2)' subst: [j] → [j] rotate: F spec: int2bits inst: name: int2bits-rec-neg

Additional information

Used terms: j