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Proof step 51 from proof bitsand-cast-01 (previous step)

# bits1 ≤ # bitsand(bits3 + bits, bits1), # bits = # bits1, # bits1 = # bitsand(bits1, bits) ⊦ lastn(# bits1, bitsand(bits3 + bits, bits1)) = bitsand(bits1, bits), bits3 = [], bits1 = [], bits3 + bits = [], bits2 = [], bits2 + bits1 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ # bits1 ≤ # bits0 ∧ bits1 ≠ [] ∧ ¬ bits1.first → bitsand(bits + bits0, bits1) = bitsand(bits0, bits1)
Proof step 53:
¬(# bits1 ≤ # bits ∧ bits1 ≠ [] ∧ ¬ bits1.first), # bits1 ≤ # bitsand(bits3 + bits, bits1), # bits = # bits1, # bits1 = # bitsand(bits1, bits) ⊦ lastn(# bits1, bitsand(bits3 + bits, bits1)) = bitsand(bits1, bits), bits3 = [], bits1 = [], bits3 + bits = [], bits2 = [], bits2 + bits1 = []
Proof step 61:
# bits1 ≤ # bits ∧ bits1 ≠ [] ∧ ¬ bits1.first, # bits1 ≤ # bitsand(bits, bits1), # bits = # bits1, # bits1 = # bitsand(bits1, bits) ⊦ lastn(# bits1, bitsand(bits, bits1)) = bitsand(bits1, bits), bits3 = [], bits1 = [], bits3 + bits = [], bits2 = [], bits2 + bits1 = []

Rule argument

name: bitsand-app-false rotate: F subst: [bits1, bits0, bits] → [bits1, bits, bits3]

Additional information

Used terms: bits1, bits, bits3