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Proof step 27 from proof uint2bits-restsgn (previous step)

uint2bits(2 * 2 ^ i - i0) = boolvar ' + bits, 0 ≤ i, 0 < i, 1 < i0, 1 < i, 0 < i0, 0 ≤ i0, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i < i0 * 2, 0 ≤ 2 ^ i, i0 * 2 < 4 * 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(bits + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = bits + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, bits = [], (¬ bits.first) ∈ bits.rest

User applied rule insert elim lemma

And obtained the following new goals

Proof step 28:
bits = boolvar0 + bits0, uint2bits(2 * 2 ^ i - i0) = boolvar ' + bits, 0 ≤ i, 0 < i, 1 < i0, 1 < i, 0 < i0, 0 ≤ i0, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i < i0 * 2, 0 ≤ 2 ^ i, i0 * 2 < 4 * 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(bits + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = bits + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, bits = [], (¬ boolvar0) ∈ bits0

Rule argument

seq: ⊦ bits ≠ [] → ( (boolvar ↔ bits.first) ∧ bits0 = bits.rest ↔ bits = boolvar + bits0) subst: [bits, boolvar, bits0] → [bits, boolvar0, bits0] rotate: T spec: list-data inst: bitlista name: elim-1

Additional information

Used simplifier rules: ⊦ bits ≠ [] → ( (boolvar ↔ bits.first) ∧ bits0 = bits.rest ↔ bits = boolvar + bits0); (elim) Used terms: bits, boolvar0, bits0