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Proof step 66 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, # int2bits(j0) ≤ i→n(i) ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, 2 ^ n→i(# fillsgn(i→n(i), bitsinvert(int2bits(j0)))) * (j + 1) = 1 + bits2uint(fillsgn(i→n(i), bitsinvert(int2bits(j0)))) + j * 2 ^ i + j0, fillsgn(i→n(i), bitsinvert(int2bits(j0))) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 67:
⊦ # bits ≤ n ∧ bits ≠ [] → fillsgn(n, bits) = mklist(bits.first, n - # bits) + bits
Proof step 68:
# bitsinvert(int2bits(j0)) ≤ 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, # int2bits(j0) ≤ i→n(i) ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, 2 ^ n→i(#(mklist(bitsinvert(int2bits(j0)).first, i→n(i) - # bitsinvert(int2bits(j0))) + bitsinvert(int2bits(j0)))) * (j + 1) = 1 + bits2uint(mklist(bitsinvert(int2bits(j0)).first, i→n(i) - # bitsinvert(int2bits(j0))) + bitsinvert(int2bits(j0))) + j * 2 ^ i + j0, mklist(bitsinvert(int2bits(j0)).first, i→n(i) - # bitsinvert(int2bits(j0))) + bitsinvert(int2bits(j0)) = []

Rule argument

seq: ⊦ # bits ≤ n ∧ bits ≠ [] → fillsgn(n, bits) = mklist(bits.first, n - # bits) + bits subst: [bits, n] → [bitsinvert(int2bits(j0)), i→n(i)] rotate: F spec: bitopsaux inst: name: fill-fill

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ # bitsinvert(bits) = # bits;% ⊦ ¬ m ≤ n ↔ n < m;%⊦ 0 ≤ j ∧ j < 2 ^ i → (i→n(i) < # int2bits(j) ↔ # int2bits(j) = i→n(i) + 1);% ⊦ bitsinvert(bits) = [] ↔ bits = [];% ⊦ int2bits(j) ≠ [];% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ m = n + 1 → ¬ m ≤ n; Used terms: bitsinvert(int2bits(j0)), i→n(i)