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Proof step 7 from proof bitsxor-append-same-size (previous step)

# bits0 = # bits2, ∀ bits1, bits, bits2, bits3. # bits3 < # bits0 → # bits3 = # bits2 → bits = [] ∨ bits3 = [] ∨ bits1 = [] ∨ bitsxor(bits + bits3, bits1 + bits2) = bitsxor(bits, bits1) + bitsxor(bits3, bits2) ⊦ # bits + # bits2 = 0, # bits1 + # bits2 = 0, bits2 = [], bits1 = [], bits0 = [], bits = [], bitsxor(bits + bits0.butlast, bits1 + bits2.butlast) + (¬(bits0.last ↔ bits2.last))' = bitsxor(bits, bits1) + bitsxor(bits0, bits2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsxor(bits, bits0) = bitsxor(bits.butlast, bits0.butlast) + (¬(bits.last ↔ bits0.last))
Proof step 9:
¬(1 < # bits0 ∧ 1 < # bits2), # bits0 = # bits2, ∀ bits1, bits, bits2, bits3. # bits3 < # bits0 → # bits3 = # bits2 → bits = [] ∨ bits3 = [] ∨ bits1 = [] ∨ bitsxor(bits + bits3, bits1 + bits2) = bitsxor(bits, bits1) + bitsxor(bits3, bits2) ⊦ # bits + # bits2 = 0, # bits1 + # bits2 = 0, bits2 = [], bits1 = [], bits0 = [], bits = [], bitsxor(bits + bits0.butlast, bits1 + bits2.butlast) + (¬(bits0.last ↔ bits2.last))' = bitsxor(bits, bits1) + bitsxor(bits0, bits2)
Proof step 14:
1 < # bits0 ∧ 1 < # bits2, # bits0 = # bits2, ∀ bits1, bits, bits2, bits3. # bits3 < # bits0 → # bits3 = # bits2 → bits = [] ∨ bits3 = [] ∨ bits1 = [] ∨ bitsxor(bits + bits3, bits1 + bits2) = bitsxor(bits, bits1) + bitsxor(bits3, bits2) ⊦ # bits + # bits2 = 0, # bits1 + # bits2 = 0, bits2 = [], bits1 = [], bits0 = [], bits = [], bitsxor(bits + bits0.butlast, bits1 + bits2.butlast) + (¬(bits0.last ↔ bits2.last))' = bitsxor(bits, bits1) + (bitsxor(bits0.butlast, bits2.butlast) + (¬(bits0.last ↔ bits2.last)))

Rule argument

name: bitsxor-rec-01 rotate: F subst: [bits0, bits] → [bits2, bits0]

Additional information

Used terms: bits2, bits0