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Proof step 140 from proof bitsand-cast (previous step)

n = # fillsgn(n, bits2 + bitsand(bits, true ' + false ' + bits1)), # bits = 2 + # bits1, n = # fillsgn(n, mklist(true, m) + true ' + false ' + bits1), sgn(mklist(true, m) + true ' + false ' + bits1) = true ' + false ' + bits1, 2 + # bits1 + m < # fillsgn(m, bits2) + # bitsand(bits, true ' + false ' + bits1), bits2.first, # bits2 < m, 2 + # bits1 + m < n, n < # fillsgn(m, bits2) + # bitsand(bits, true ' + false ' + bits1), 3 + # bits1 + m < # fillsgn(m, bits2) + # bitsand(bits, true ' + false ' + bits1) ⊦ bits2 = [], bits = [], bits2 + bits = [], fillsgn(n, bits2 + bitsand(bits, true ' + false ' + bits1)) = lastn(n, fillsgn(n, fillsgn(m, bits2) + bitsand(bits, true ' + false ' + bits1))), n = # fillsgn(n, fillsgn(m, bits2) + bitsand(bits, true ' + false ' + bits1))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 141:
⊦ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits
Proof step 142:
(true ' + false ' + bits1).first ∧ #(true ' + false ' + bits1) ≤ # bits ∧ true ' + false ' + bits1 ≠ [], n = # fillsgn(n, bits2 + bitsand(bits, true ' + false ' + bits1)), # bits = 2 + # bits1, n = # fillsgn(n, mklist(true, m) + true ' + false ' + bits1), sgn(mklist(true, m) + true ' + false ' + bits1) = true ' + false ' + bits1, 2 + # bits1 + m < # fillsgn(m, bits2) + # bits, bits2.first, # bits2 < m, 2 + # bits1 + m < n, n < # fillsgn(m, bits2) + # bits, 3 + # bits1 + m < # fillsgn(m, bits2) + # bits ⊦ bits2 = [], bits = [], bits2 + bits = [], fillsgn(n, bits2 + bitsand(bits, true ' + false ' + bits1)) = lastn(n, fillsgn(n, fillsgn(m, bits2) + bitsand(bits, true ' + false ' + bits1))), n = # fillsgn(n, fillsgn(m, bits2) + bitsand(bits, true ' + false ' + bits1))

Rule argument

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

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 ' + bits1, bits