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Proof step 4 from proof int2bits-neg-03 (previous step)

j < 0 → bits2int(int2bits(j)) = bits2int(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest), j < 0 ⊦ int2bits(j) = sgn(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ bits ≠ [] ∧ bits0 ≠ [] → (bits2int(bits) = bits2int(bits0) ↔ sgn(bits) = sgn(bits0))
Proof step 6:
¬(int2bits(j) ≠ [] ∧ int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest ≠ []), j < 0 → bits2int(int2bits(j)) = bits2int(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest), j < 0 ⊦ int2bits(j) = sgn(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest)

Rule argument

name: bits2int-inj rotate: F subst: [bits0, bits] → [int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest, int2bits(j)]

Additional information

Used simplifier rules: int2bits-not-empty: ⊦ int2bits(j) ≠ [];%int2bits-sgn: ⊦ sgn(int2bits(i)) = int2bits(i);% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest, int2bits(j)