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Proof step 16 from proof bitsand-app-zeros (previous step)

# bits1 < # bits, ∀ bits0, bits, bits2. # bits2 < # bits1 → true → sgn(bitsand(bits + false ' + bits0, false ' + bits2)) = sgn(bitsand(false ' + bits0, false ' + bits2)) ∨ true ∈ bits ⊦ false ' + bits1 = true ', bits1 = [], bits = [], sgn(bitsand(bits, false ' + bits1.butlast) + false ') = false ', true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: bitsand-zeros-01 rotate: F subst: [bits0, bits] → [false ' + bits1.butlast, bits]

Additional information

Used simplifier rules: ⊦ boolvar ' = boolvar0 ' ↔ boolvar ↔ boolvar0;% ⊦ sgn(false ' + false ') = false ';% ⊦ sgn(false ' + mklist(false, n) + bits) = sgn(false ' + bits);% ⊦ bits + [] = bits;% ⊦ mklist(boolvar, 0) = [];%n ≠ 0 ⊦ mklist(false, (* n)) = false ' + mklist(false, n -1);% ⊦ mklist(boolvar, m + n) = mklist(boolvar, m) + mklist(boolvar, n);% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ bits ≠ [] → # bits.butlast = # bits - 1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ (boolvar ' + bits).first ↔ boolvar;% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ m < n + 1 ↔ ¬ n < m;% ⊦ # bits = 0 ↔ bits = [];% ⊦ m < 1 ↔ m = 0;% ⊦ ¬ m ≤ n ↔ n < m;%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0;% ⊦ ¬ n < n;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used terms: false ' + bits1.butlast, bits