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Proof step 670 from proof bitsand-cast (previous step)

2 + # bits = # bits1, # bits1 = # bitsand(bits1, false ' + true ' + bits), # bits3 + # bitsand(false ' + true ' + bits, bits1) < 2 + # bits + m, # bits3 < m ⊦ bits3 = [], bits3 + bits1 = [], bits2 + mklist(false, m) = [], bits1 = [], mklist(false, m) + bitsand(bits1, false ' + true ' + bits) = lastn(2 + # bits + m, fillsgn(2 + # bits + m, mklist(false, # bits3) + bitsand(false ' + true ' + bits, bits1))), # bits3 = m, bits3.first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 671:
⊦ ¬ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits0
Proof step 672:
¬ (false ' + true ' + bits).first ∧ #(false ' + true ' + bits) ≤ # bits1 ∧ false ' + true ' + bits ≠ [], 2 + # bits = # bits1, # bits1 = #(false ' + true ' + bits), # bits3 + # bitsand(false ' + true ' + bits, bits1) < 2 + # bits + m, # bits3 < m ⊦ bits3 = [], bits3 + bits1 = [], bits2 + mklist(false, m) = [], bits1 = [], mklist(false, m) + bitsand(bits1, false ' + true ' + bits) = lastn(2 + # bits + m, fillsgn(2 + # bits + m, mklist(false, # bits3) + bitsand(false ' + true ' + bits, bits1))), # bits3 = m, bits3.first

Rule argument

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