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Proof step 30 from proof bor-2pot-plus-01 (previous step)

0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j + 1 < 2 ^ i, j < j0, j < i, 0 < 2 ^ i, 0 ≤ j0, j + 1 < i, j < 0 ⊦ i = 0, bits2int(bitsinvert(bitsand(bitsinvert(int2bits(j)) + mklist(true, i→n(i)), bitsinvert(int2bits(j0))))) = j * 2 ^ i + j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 32:
bitsinvert(int2bits(j)) + mklist(true, i→n(i)) ≠ [] ∧ bitsinvert(int2bits(j0)) ≠ [], 0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j + 1 < 2 ^ i, j < j0, j < i, 0 < 2 ^ i, 0 ≤ j0, j + 1 < i, j < 0 ⊦ i = 0, bits2int(bitsinvert(bitsand(bitsinvert(int2bits(j0)), bitsinvert(int2bits(j)) + mklist(true, i→n(i))))) = j * 2 ^ i + j0

Rule argument

name: and-com rotate: F subst: [bits0, bits] → [bitsinvert(int2bits(j0)), bitsinvert(int2bits(j)) + mklist(true, i→n(i))]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ bitsinvert(bits) = [] ↔ bits = [];% ⊦ int2bits(j) ≠ [];% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: bitsinvert(int2bits(j0)), bitsinvert(int2bits(j)) + mklist(true, i→n(i))