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Proof step 72 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) ⊦ mklist(true, i→n(i) - # int2bits(j0)) + bitsinvert(int2bits(j0)) = [], bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, (j + 1) * 2 ^ (n→i(i→n(i) - # int2bits(j0)) + n→i(# int2bits(j0))) = 2 ^ n→i(# int2bits(j0)) * (bits2uint(mklist(true, i→n(i) - # int2bits(j0))) + 1) + j * 2 ^ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 73:
⊦ ¬ false ∈ bits ∧ bits ≠ [] → bits2uint(bits) = 2 ^ n→i(# bits) - 1
Proof step 74:
¬ false ∈ mklist(true, i→n(i) - # int2bits(j0)) ∧ mklist(true, i→n(i) - # 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) ⊦ mklist(true, i→n(i) - # int2bits(j0)) + bitsinvert(int2bits(j0)) = [], bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, (j + 1) * 2 ^ (n→i(i→n(i) - # int2bits(j0)) + n→i(# int2bits(j0))) = 2 ^ n→i(# int2bits(j0)) * ((2 ^ n→i(# mklist(true, i→n(i) - # int2bits(j0))) - 1) + 1) + j * 2 ^ i

Rule argument

seq: ⊦ ¬ false ∈ bits ∧ bits ≠ [] → bits2uint(bits) = 2 ^ n→i(# bits) - 1 subst: [bits] → [mklist(true, i→n(i) - # int2bits(j0))] rotate: F spec: int2bits inst: name: bits2uint-ones

Additional information

Used simplifier rules: ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ¬ false ∈ mklist(true, n);% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ m ≤ n → (n - m = 0 ↔ n = m);% ⊦ 0 < i → n→i(i→n(i)) = i;% ⊦ m - m = 0;% ⊦ i + 0 = i;% ⊦ mklist(boolvar, 0) = [];% ⊦ bits2uint([]) = 0;% ⊦ 1 * i = i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i * j + i = i * (j + 1);% ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); Used terms: mklist(true, i→n(i) - # int2bits(j0))