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Proof step 20 from proof band-min-02 (previous step)

j < 0, i < 0, i0 + 2 ^ (n→i(# int2bits(j)) - 1) ≤ 0, i0 + 2 ^ (n→i(# int2bits(i)) - 1) ≤ 0, bits2int(bitsand(int2bits(j), int2bits(i))) < i0, # int2bits(i) < # int2bits(j), ¬ (# int2bits(j) - 1) + 1 < # bitsand(int2bits(j), int2bits(i)) → 0 ≤ 2 ^ n→i(# int2bits(j) - 1) + bits2int(bitsand(int2bits(j), int2bits(i))) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits
Proof step 22:
int2bits(i).first ∧ # int2bits(i) ≤ # int2bits(j) ∧ int2bits(i) ≠ [], j < 0, i < 0, i0 + 2 ^ (n→i(# int2bits(j)) - 1) ≤ 0, i0 + 2 ^ (n→i(# int2bits(i)) - 1) ≤ 0, bits2int(bitsand(int2bits(j), int2bits(i))) < i0, # int2bits(i) < # int2bits(j), ¬ (# int2bits(j) - 1) + 1 < # int2bits(j) → 0 ≤ 2 ^ n→i(# int2bits(j) - 1) + bits2int(bitsand(int2bits(j), int2bits(i))) ⊦

Rule argument

name: bitsand-len-true rotate: F subst: [bits0, bits] → [int2bits(i), int2bits(j)]

Additional information

Used simplifier rules: ⊦ j < 0 → int2bits(j).first;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m < n → ¬ n < m;% ⊦ int2bits(j) ≠ []; Used terms: int2bits(i), int2bits(j)