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Proof step 63 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, (j + 1) * 2 ^ n→i(# bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))) = (j * 2 ^ i + j0) + 1 + bits2uint(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))), bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i))) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
⊦ ¬ false ∈ bits ∧ # bits0 ≤ # bits ∧ bits0.first ∧ bits0 ≠ [] → bitsand(bits0, bits) = fillsgn(# bits, bits0)
Proof step 65:
¬ false ∈ mklist(true, i→n(i)) ∧ # bitsinvert(int2bits(j0)) ≤ # mklist(true, i→n(i)) ∧ bitsinvert(int2bits(j0)).first ∧ 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, (j + 1) * 2 ^ n→i(# fillsgn(# mklist(true, i→n(i)), bitsinvert(int2bits(j0)))) = (j * 2 ^ i + j0) + 1 + bits2uint(fillsgn(# mklist(true, i→n(i)), bitsinvert(int2bits(j0)))), fillsgn(# mklist(true, i→n(i)), bitsinvert(int2bits(j0))) = []

Rule argument

name: add-ones-grlen-01 rotate: F subst: [bits, bits0] → [mklist(true, i→n(i)), bitsinvert(int2bits(j0))]

Additional information

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