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Proof step 33 from proof band-011x-pos (previous step)

0 < j, 0 ≤ j, 0 < i, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ j = 1, i = 0, bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j / 2) + (j % 2 = 1)')) = j % 2 ^ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 35:
1 < #(false ' + mklist(true, i→n(i))) ∧ 1 < #(uint2bits(j / 2) + (j % 2 = 1)'), 0 < j, 0 ≤ j, 0 < i, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ j0 ∧ 0 ≤ j0 ∧ 0 ≤ i → bits2int(bitsand(false ' + mklist(true, i→n(i)), uint2bits(j0))) = j0 % 2 ^ i ⊦ j = 1, i = 0, bits2int ( bitsand((false ' + mklist(true, i→n(i))).butlast, (uint2bits(j / 2) + (j % 2 = 1)').butlast) + ((false ' + mklist(true, i→n(i))).last ∧ (uint2bits(j / 2) + (j % 2 = 1)').last)) = j % 2 ^ i

Rule argument

name: and-rec rotate: F subst: [bits0, bits] → [uint2bits(j / 2) + (j % 2 = 1)', false ' + mklist(true, i→n(i))]

Additional information

Used simplifier rules: ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ # mklist(boolvar, n) = n;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ m + 1 ≠ 0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ 1 = m + 1 ↔ m = 0;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ # bits = 0 ↔ bits = [];% ⊦ i < j ∧ ¬ i0 < j → ¬ i0 < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i;%0 ≤ i ⊦ uint2bits(i) ≠ []; Used terms: uint2bits(j / 2) + (j % 2 = 1)', false ' + mklist(true, i→n(i))