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Proof step 29 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 ⊦ bits4 = [], sgn(bitsand(bits + false ' + bits3, bits4) + (boolvar ∧ boolvar0)') = sgn(bitsand(false ' + bits3, bits4) + (boolvar ∧ boolvar0)'), true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ true ∈ bits ∧ false ∈ bits → sgn(bits + boolvar ') = sgn(bits) + boolvar '
Proof step 31:
¬(true ∈ bitsand(false ' + bits3, bits4) ∧ false ∈ bitsand(false ' + bits3, bits4)), ∀ bits0, bits, bits2. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ bits4 = [], sgn(bitsand(bits + false ' + bits3, bits4) + (boolvar ∧ boolvar0)') = sgn(bitsand(false ' + bits3, bits4) + (boolvar ∧ boolvar0)'), true ∈ bits
Proof step 42:
true ∈ bitsand(false ' + bits3, bits4) ∧ false ∈ bitsand(false ' + bits3, bits4), ∀ bits0, bits, bits2. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ bits4 = [], sgn(bitsand(bits + false ' + bits3, bits4) + (boolvar ∧ boolvar0)') = sgn(bitsand(false ' + bits3, bits4)) + (boolvar ∧ boolvar0)', true ∈ bits

Rule argument

seq: ⊦ true ∈ bits ∧ false ∈ bits → sgn(bits + boolvar ') = sgn(bits) + boolvar ' subst: [boolvar, bits] → [boolvar ∧ boolvar0, bitsand(false ' + bits3, bits4)] rotate: F spec: bitopsaux inst: name: sgn-last-both

Additional information

Used terms: boolvar ∧ boolvar0, bitsand(false ' + bits3, bits4)