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Proof step 15 from proof bitsand-app-zeros (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
# 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

Rule argument

Additional information

Used simplifier rules: and-rec-06: ⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits0 + boolvar0 ', boolvar1 ' + bits) = bitsand(bits0, boolvar1 ' + bits.butlast) + (boolvar0 ∧ bits.last);% ⊦ bits + boolvar = bits + boolvar ';% ⊦ # bits = 0 ↔ bits = [];% ⊦ m < n → n ≠ 0; (forward)% ⊦ boolvar ' ≠ [];% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ (boolvar ' + bits).first ↔ boolvar;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m + n ≤ m + n0 ↔ n ≤ n0;% ⊦ # boolvar ' = 0 +1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ boolvar ∈ boolvar0 ' ↔ boolvar ↔ boolvar0;% ⊦ boolvar ∈ bits + bits0 ↔ ¬ ¬(boolvar ∈ bits ∨ boolvar ∈ bits0);