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Proof step 564 from proof bitsand-cast (previous step)

sgn(mklist(true, # bits0) + true ' + false ' + bits) = true ' + false ' + bits, # bits1 = 2 + # bits, # bits1 < # bitsand(bits1, true ' + false ' + bits), mklist(true, # bits0).first, 2 + # bits + # bits0 ≤ # bitsand(bits0, mklist(true, # bits0)) + # bitsand(bits1, true ' + false ' + bits), # bits1 ≤ # bits4 + # bitsand(bits1, true ' + false ' + bits) ⊦ bits0 = [], bits4 = [], bits2 = [], (bits4 + bits0) + bits1 = [], bits1 = [], lastn(2 + # bits + # bits0, bits0 + bitsand(bits1, true ' + false ' + bits)) = lastn(2 + # bits + # bits0, bitsand(bits0, mklist(true, # bits0)) + bitsand(bits1, true ' + false ' + bits)), bits2 + mklist(true, # bits0) = [], bits4 + bits0 = [], # bits1 = # bitsand(bits1, true ' + false ' + bits)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 565:
⊦ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits

Rule argument

name: bitsand-len-true rotate: F subst: [bits0, bits] → [true ' + false ' + bits, bits1]

Additional information

Used simplifier rules: ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ (boolvar ' + bits).first ↔ boolvar;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ n < n0 ∧ n0 < n1 → n +1 < n1; (forward)% ⊦ m < n ∧ m = n0 → n0 < n; (forward)% ⊦ m < n ∧ n = n0 → m < n0; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward)% ⊦ # bits = 0 ↔ bits = [];% ⊦ m +1 = m + 1;% ⊦ m < n → ¬ n < m; Used terms: true ' + false ' + bits, bits1