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Proof step 13 from proof bitsxor-is-orand (previous step)

# bits = 1, ∀ bits, bits1. # bits1 < # bits0 → true → bits = [] ∨ bits1 = [] ∨ bits2int(bitsxor(bits, bits1)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits1), bitsand(bitsinvert(bits1), bits))) ⊦ bits2int(bitsxor(bits, bits0)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits0), bitsand(bitsinvert(bits0), bits))), bits0 = [], bits = [], bits0 = true ', bits0 = false ', # bits0 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ bits0 ≠ [] ∧ bits = true ' → bitsand(bits, bits0) = bits0
Proof step 15:
¬(bits0 ≠ [] ∧ bitsinvert(bits) = true '), # bits = 1, ∀ bits, bits1. # bits1 < # bits0 → true → bits = [] ∨ bits1 = [] ∨ bits2int(bitsxor(bits, bits1)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits1), bitsand(bitsinvert(bits1), bits))) ⊦ bits2int(bitsxor(bits, bits0)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits0), bitsand(bitsinvert(bits0), bits))), bits0 = [], bits = [], bits0 = true ', bits0 = false ', # bits0 = 1
Proof step 24:
bits0 ≠ [] ∧ bitsinvert(bits) = true ', # bits = 1, ∀ bits, bits1. # bits1 < # bits0 → true → bits = [] ∨ bits1 = [] ∨ bits2int(bitsxor(bits, bits1)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits1), bitsand(bitsinvert(bits1), bits))) ⊦ bits2int(bitsxor(bits, bits0)) = bits2int(bitsor(bits0, bitsand(bitsinvert(bits0), bits))), bits0 = [], bits = [], bits0 = true ', bits0 = false ', # bits0 = 1

Rule argument

name: and-rec-03 rotate: F subst: [bits0, bits] → [bits0, bitsinvert(bits)]

Additional information

Used terms: bits0, bitsinvert(bits)