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Proof step 36 from proof bor-2pot-plus-01 (previous step)

# int2bits(j0) = i→n(i) + 1, 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, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ bits.first ∧ # bits = # bits1 + 1 ∧ bits1 ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0 + bits1) = bits0 + bitsand(bits.rest, bits1)
Proof step 38:
bitsinvert(int2bits(j0)).first ∧ # bitsinvert(int2bits(j0)) = # mklist(true, i→n(i)) + 1 ∧ mklist(true, i→n(i)) ≠ [] ∧ bitsinvert(int2bits(j)) ≠ [], # int2bits(j0) = i→n(i) + 1, 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(bitsinvert(int2bits(j)) + bitsand(bitsinvert(int2bits(j0)).rest, mklist(true, i→n(i))))) = j * 2 ^ i + j0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = []

Rule argument

name: and-size-one-01 rotate: F subst: [bits1, bits0, bits] → [mklist(true, i→n(i)), bitsinvert(int2bits(j)), bitsinvert(int2bits(j0))]

Additional information

Used simplifier rules: ⊦ m + n = n + m;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 ≤ j → bitsinvert(int2bits(j)).first;% ⊦ # bitsinvert(bits) = # bits;% ⊦ # mklist(boolvar, n) = n;% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ bitsinvert(bits) = [] ↔ bits = [];% ⊦ int2bits(j) ≠ []; Used terms: mklist(true, i→n(i)), bitsinvert(int2bits(j)), bitsinvert(int2bits(j0))