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Proof step 10 from proof bitsand-app-zeros-01 (previous step)

∀ bits0, bits, bits2. # bits2 < # bits1 → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ sgn(bitsand(bits + false ' + bits0, bits1)) = sgn(bitsand(false ' + bits0, bits1)), bits1 = [], bits1 = true ', true ∈ bits, bits0 ≠ [] ∧ # bits1 ≠ 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ bits ≠ [] ∧ bits0 = false ' → bitsand(bits, bits0) = false '
Proof step 12:
¬(false ' + bits0 ≠ [] ∧ bits1 = false '), ∀ bits0, bits, bits2. # bits2 < # bits1 → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ sgn(bitsand(bits + false ' + bits0, bits1)) = sgn(bitsand(false ' + bits0, bits1)), bits1 = [], bits1 = true ', true ∈ bits, bits0 ≠ [] ∧ # bits1 ≠ 1
Proof step 15:
false ' + bits0 ≠ [] ∧ bits1 = false ', ∀ bits0, bits, bits2. # bits2 < # bits1 → true → bits2 = [] ∨ sgn(bitsand(bits + false ' + bits0, bits2)) = sgn(bitsand(false ' + bits0, bits2)) ∨ true ∈ bits ⊦ sgn(bitsand(bits + false ' + bits0, bits1)) = sgn(false '), bits1 = [], bits1 = true ', true ∈ bits, bits0 ≠ [] ∧ # bits1 ≠ 1

Rule argument

name: and-rec-02 rotate: F subst: [bits0, bits] → [bits1, false ' + bits0]

Additional information

Used terms: bits1, false ' + bits0