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Proof step 77 from proof band-min (previous step)

# int2bits(i2 / 2) = # int2bits(i / 2), i2 < ~1, i2 < 0, i < ~1, 2 ^ (n→i(# int2bits(i / 2)) + 1) + i0 ≤ 0, i < 0, 2 * bits2int(bitsand(int2bits(i2 / 2), int2bits(i / 2))) < i0, ∀ 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, ~2 = i, ~1 = (i2 + 1) % 2 ∧ ~1 = (i + 1) % 2

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 < i ∧ j < 0 → ((* i) * i0 < (* j) ↔ i0 < j / i) subst: [j, i0, i] → [i0, bits2int(bitsand(int2bits(i2 / 2), int2bits(i / 2))), 2] rotate: F spec: int-div inst: name: nummult-ls-01

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i < 0 ∧ 0 ≤ j → i < j;% ⊦ i + j ≤ i ↔ j ≤ 0;% ⊦ 2 ^ i ≤ 0 ↔ i < 0;% ⊦ ¬ n→i(n) + 1 < 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i; Used terms: i0, bits2int(bitsand(int2bits(i2 / 2), int2bits(i / 2))), 2