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Proof step 35 from proof and-sgn (previous step)

∀ bits1, bits. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ sgn(bitsand(sgn(bits), bits1)) = sgn(bitsand(bits, bits1)) ⊦ bits2 + boolvar ' = (bits2 + boolvar ').first ', bits0 = [], sgn(bitsand(sgn(bits0 + boolvar0 '), bits2 + boolvar ')) = sgn(bitsand(bits0, bits2) + (boolvar0 ∧ boolvar)'), bits0 + boolvar0 ' = bits0.first '

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
(¬ boolvar) ∈ bits ⊦ sgn(bits + boolvar ') = sgn(bits) + boolvar '
Proof step 37:
¬ (¬ boolvar0) ∈ bits0, ∀ bits1, bits. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ sgn(bitsand(sgn(bits), bits1)) = sgn(bitsand(bits, bits1)) ⊦ bits2 + boolvar ' = (bits2 + boolvar ').first ', bits0 = [], sgn(bitsand(sgn(bits0 + boolvar0 '), bits2 + boolvar ')) = sgn(bitsand(bits0, bits2) + (boolvar0 ∧ boolvar)'), bits0 + boolvar0 ' = bits0.first '
Proof step 68:
(¬ boolvar0) ∈ bits0, ∀ bits1, bits. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ sgn(bitsand(sgn(bits), bits1)) = sgn(bitsand(bits, bits1)) ⊦ bits2 + boolvar ' = (bits2 + boolvar ').first ', bits0 = [], sgn(bitsand(sgn(bits0) + boolvar0 ', bits2 + boolvar ')) = sgn(bitsand(bits0, bits2) + (boolvar0 ∧ boolvar)'), bits0 + boolvar0 ' = bits0.first '

Rule argument

seq: (¬ boolvar) ∈ bits ⊦ sgn(bits + boolvar ') = sgn(bits) + boolvar ' subst: [boolvar, bits] → [boolvar0, bits0] rotate: F spec: bitopsaux inst: name: sgn-last-diff

Additional information

Used terms: boolvar0, bits0