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Proof step 52 from proof bitsand-fillsgn (previous step)

∀ n, bits, bits1. # bits1 < #(false + bits2) → true → bits = [] ∨ bits1 = [] ∨ fillsgn(n, bitsand(fillsgn(n, bits), bits1)) = fillsgn(n, bitsand(bits, bits1)) ⊦ fillsgn(n0 + # bits2, bitsand(mklist(true, n0) + mklist(true, # bits2), false ' + bits2)) = mklist(false, n0) + bits2, n0 = 0, n0 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 53:
⊦ # bits = # bits1 ∧ bits0 ≠ [] ∧ bits2 ≠ [] ∧ bits ≠ [] → bitsand(bits0 + bits, bits2 + bits1) = bitsand(bits0, bits2) + bitsand(bits, bits1)
Proof step 54:
¬(# mklist(true, # bits2) = # bits2 ∧ mklist(true, n0) ≠ [] ∧ false ' ≠ [] ∧ mklist(true, # bits2) ≠ []), ∀ n, bits, bits1. # bits1 < #(false + bits2) → true → bits = [] ∨ bits1 = [] ∨ fillsgn(n, bitsand(fillsgn(n, bits), bits1)) = fillsgn(n, bitsand(bits, bits1)) ⊦ fillsgn(n0 + # bits2, bitsand(mklist(true, n0) + mklist(true, # bits2), false ' + bits2)) = mklist(false, n0) + bits2, n0 = 0, n0 = 1
Proof step 60:
# mklist(true, # bits2) = # bits2 ∧ mklist(true, n0) ≠ [] ∧ false ' ≠ [] ∧ mklist(true, # bits2) ≠ [], ∀ n, bits, bits1. # bits1 < #(false + bits2) → true → bits = [] ∨ bits1 = [] ∨ fillsgn(n, bitsand(fillsgn(n, bits), bits1)) = fillsgn(n, bitsand(bits, bits1)) ⊦ fillsgn(n0 + # bits2, bitsand(mklist(true, n0), false ') + bitsand(mklist(true, # bits2), bits2)) = mklist(false, n0) + bits2, n0 = 0, n0 = 1

Rule argument

name: bitsand-app rotate: F subst: [bits1, bits2, bits, bits0] → [bits2, false ', mklist(true, # bits2), mklist(true, n0)]

Additional information

Used terms: bits2, false ', mklist(true, # bits2), mklist(true, n0)