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Proof step 429 from proof bitsand-cast (previous step)

# bits1 = 2 + # bits, sgn((mklist(true, n) + mklist(true, # bits3)) + true ' + false ' + bits) = true ' + false ' + bits, bits3.first, # bitsand(bits1, true ' + false ' + bits) ≤ 2 + # bits + n ⊦ bits1 = [], bits3 + bits1 = [], bits3 = [], bits2 + mklist(true, n) + mklist(true, # bits3) = [], bits3 + bitsand(bits1, true ' + false ' + bits) = [], lastn (2 + # bits + n + # bits3, mklist(true, 2 + # bits + n - # bitsand(bits1, true ' + false ' + bits)) + bits3 + bitsand(bits1, true ' + false ' + bits)) = lastn(2 + # bits + n + # bits3, fillsgn(2 + # bits + n + # bits3, bits2 + mklist(true, n) + bits3 + bitsand(bits1, true ' + false ' + bits)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 430:
⊦ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits
Proof step 431:
(true ' + false ' + bits).first ∧ #(true ' + false ' + bits) ≤ # bits1 ∧ true ' + false ' + bits ≠ [], # bits1 = 2 + # bits, sgn((mklist(true, n) + mklist(true, # bits3)) + true ' + false ' + bits) = true ' + false ' + bits, bits3.first, # bits1 ≤ 2 + # bits + n ⊦ bits1 = [], bits3 + bits1 = [], bits3 = [], bits2 + mklist(true, n) + mklist(true, # bits3) = [], bits3 + bitsand(bits1, true ' + false ' + bits) = [], lastn(2 + # bits + n + # bits3, mklist(true, 2 + # bits + n - # bits1) + bits3 + bitsand(bits1, true ' + false ' + bits)) = lastn(2 + # bits + n + # bits3, fillsgn(2 + # bits + n + # bits3, bits2 + mklist(true, n) + bits3 + bitsand(bits1, true ' + false ' + bits)))

Rule argument

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

Additional information

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