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Proof step 69 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(i→n(i) - # int2bits(j0)) + n→i(# int2bits(j0))) = 1 + j * 2 ^ i + bits2uint(mklist(true, i→n(i) - # int2bits(j0)) + bitsinvert(int2bits(j0))) + j0, mklist(true, i→n(i) - # int2bits(j0)) + bitsinvert(int2bits(j0)) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ # bits0 ≠ 0 → bits2uint(bits + bits0) = bits2uint(bits) * 2 ^ n→i(# bits0) + bits2uint(bits0)
Proof step 71:
# bitsinvert(int2bits(j0)) ≠ 0, 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(i→n(i) - # int2bits(j0)) + n→i(# int2bits(j0))) = 1 + j * 2 ^ i + (bits2uint(mklist(true, i→n(i) - # int2bits(j0))) * 2 ^ n→i(# bitsinvert(int2bits(j0))) + bits2uint(bitsinvert(int2bits(j0)))) + j0, mklist(true, i→n(i) - # int2bits(j0)) + bitsinvert(int2bits(j0)) = []

Rule argument

seq: ⊦ # bits0 ≠ 0 → bits2uint(bits + bits0) = bits2uint(bits) * 2 ^ n→i(# bits0) + bits2uint(bits0) subst: [bits0, bits] → [bitsinvert(int2bits(j0)), mklist(true, i→n(i) - # int2bits(j0))] rotate: F spec: int2bits inst: name: bits2uint-split

Additional information

Used simplifier rules: ⊦ # bitsinvert(bits) = # bits;% ⊦ # bits = 0 ↔ bits = [];% ⊦ int2bits(j) ≠ []; Used terms: bitsinvert(int2bits(j0)), mklist(true, i→n(i) - # int2bits(j0))