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Proof step 26 from proof bitsand-app-zeros-01 (previous step)

∀ bits0, bits, bits2. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ sgn(bitsand(bits + false ' + bits3 + boolvar ', bits4 + boolvar0 ')) = sgn(bitsand(false ' + bits3, bits4) + (boolvar ∧ boolvar0)'), bits4 = [], true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 28:
1 < #(bits + false ' + bits3 + boolvar ') ∧ 1 < #(bits4 + boolvar0 '), ∀ bits0, bits, bits2. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ sgn ( bitsand((bits + false ' + bits3 + boolvar ').butlast, (bits4 + boolvar0 ').butlast) + ((bits + false ' + bits3 + boolvar ').last ∧ (bits4 + boolvar0 ').last)) = sgn(bitsand(false ' + bits3, bits4) + (boolvar ∧ boolvar0)'), bits4 = [], true ∈ bits

Rule argument

name: and-rec rotate: F subst: [bits0, bits] → [bits4 + boolvar0 ', bits + false ' + bits3 + boolvar ']

Additional information

Used simplifier rules: ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0;% ⊦ # bits = 0 ↔ bits = [];% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;%n < n0 ⊦ (* n) ≠ m0 + (* n0);% ⊦ m + 1 ≠ 0;% ⊦ 1 = m + 1 ↔ m = 0; Used terms: bits4 + boolvar0 ', bits + false ' + bits3 + boolvar '