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Proof step 50 from proof and-sgn (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 51:
(¬ boolvar) ∈ bits ⊦ sgn(bits + boolvar ') = sgn(bits) + boolvar '
Proof step 52:
¬ (¬ boolvar) ∈ bitsand(bits0, bits2), ∀ bits1, bits. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ sgn(bitsand(sgn(bits), bits1)) = sgn(bitsand(bits, bits1)) ⊦ bits0 = [], bits2 + boolvar ' = (bits2 + boolvar ').first ', bits0 + true ' = bits0.first ', sgn(bits2 + boolvar ') = sgn(bitsand(bits0, bits2) + boolvar '), false ∈ bits0
Proof step 61:
(¬ boolvar) ∈ bitsand(bits0, bits2), ∀ bits1, bits. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ sgn(bitsand(sgn(bits), bits1)) = sgn(bitsand(bits, bits1)) ⊦ bits0 = [], bits2 + boolvar ' = (bits2 + boolvar ').first ', bits0 + true ' = bits0.first ', sgn(bits2 + boolvar ') = sgn(bitsand(bits0, bits2)) + boolvar ', false ∈ bits0

Rule argument

seq: (¬ boolvar) ∈ bits ⊦ sgn(bits + boolvar ') = sgn(bits) + boolvar ' subst: [boolvar, bits] → [boolvar, bitsand(bits0, bits2)] rotate: F spec: bitopsaux inst: name: sgn-last-diff

Additional information

Used terms: boolvar, bitsand(bits0, bits2)