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Proof step 37 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, bits4) + (boolvar ∧ boolvar0)') = sgn(bitsand(false ' + bits3, bits4) + (boolvar ∧ boolvar0)'), bits4 = [], true ∈ bitsand(false ' + bits3, bits4), true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ ¬ true ∈ bits ∧ bits ≠ [] → sgn(bits + bits0) = sgn(false ' + bits0)
Proof step 39:
¬ true ∈ bitsand(false ' + bits3, bits4) ∧ 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 ⊦ sgn(bitsand(bits + false ' + bits3, bits4) + (boolvar ∧ boolvar0)') = sgn(false ' + (boolvar ∧ boolvar0)'), bits4 = [], true ∈ bitsand(false ' + bits3, bits4), true ∈ bits

Rule argument

seq: ⊦ ¬ true ∈ bits ∧ bits ≠ [] → sgn(bits + bits0) = sgn(false ' + bits0) subst: [bits0, bits] → [(boolvar ∧ boolvar0)', bitsand(false ' + bits3, bits4)] rotate: F spec: bitopsaux inst: name: sgn-append-04

Additional information

Used simplifier rules: and-notempty-02: ⊦ bits0 ≠ [] → bitsand(boolvar ' + bits, bits0) ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: (boolvar ∧ boolvar0)', bitsand(false ' + bits3, bits4)