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Proof step 45 from proof bor-2pot-plus-01 (previous step)

# int2bits(j0) = i→n(i) + 1, j < 0, j + 1 < i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 < i, i→n(i) ≤ # int2bits(j0).rest ⊦ i = 0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], bits2int(int2bits(j) + int2bits(j0).rest) = j * 2 ^ i + j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ # bits0 ≠ 0 ∧ # bits ≠ 0 ∧ bits.first → bits2int(bits + bits0) = ~(2 ^ n→i(# bits0) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(bits0)) + 1
Proof step 47:
# int2bits(j0).rest ≠ 0 ∧ # int2bits(j) ≠ 0 ∧ int2bits(j).first, # int2bits(j0) = i→n(i) + 1, j < 0, j + 1 < i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 < i, i→n(i) ≤ # int2bits(j0).rest ⊦ i = 0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], ~(2 ^ n→i(# int2bits(j0).rest) * bits2uint(bitsinvert(int2bits(j)))) - bits2uint(bitsinvert(int2bits(j0).rest)) + 1 = j * 2 ^ i + j0

Rule argument

seq: ⊦ # bits0 ≠ 0 ∧ # bits ≠ 0 ∧ bits.first → bits2int(bits + bits0) = ~(2 ^ n→i(# bits0) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(bits0)) + 1 subst: [bits0, bits] → [int2bits(j0).rest, int2bits(j)] rotate: F spec: int2bits inst: name: bits2int-split-01

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ # bits = 0 ↔ bits = [];% ⊦ int2bits(j) ≠ [];% ⊦ j < 0 → int2bits(j).first;% ⊦ bits + [] = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ bits2int(int2bits(i)) = i;% ⊦ # [] = 0;% ⊦ i ≠ j → (i ≤ j ↔ i < j);%0 ≤ i ⊦ i→n(i) ≤ (* m) ↔ i ≤ n→i(m);% ⊦ i < j → ¬ j < i; Used terms: int2bits(j0).rest, int2bits(j)