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Proof step 127 from proof uint2bits-restsgn (previous step)

0 ≤ i, 2 ^ i0 = i2 + 1, i + 1 = i0 + n→i(# bits), ~1 ≤ i2, 0 ≤ i, (2 * 2 ^ n→i(# bits)) * i2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ (2 * 2 ^ n→i(# bits)) * i2, 2 ^ n→i(# bits) * i2 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, 2 ^ n→i(# bits) * i2 < 2 ^ (i + 1), 0 < i2, ∀ j0, i1. i→n(j0) < i→n((2 * 2 ^ n→i(# bits)) * i2 + 1) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * (2 ^ n→i(# bits) * i2 + 1), false ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 128:
⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j
Proof step 129:
0 ≤ n→i(# bits), 0 ≤ i, 2 ^ i0 = i2 + 1, i + 1 = i0 + n→i(# bits), ~1 ≤ i2, 0 ≤ i, 2 ^ (n→i(# bits) + 1) * i2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ 2 ^ (n→i(# bits) + 1) * i2, 2 ^ n→i(# bits) * i2 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, 2 ^ n→i(# bits) * i2 < 2 ^ (i + 1), 0 < i2, ∀ j0, i1. i→n(j0) < i→n(2 ^ (n→i(# bits) + 1) * i2 + 1) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * (2 ^ n→i(# bits) * i2 + 1), false ∈ bits

Rule argument

seq: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j subst: [i, j] → [n→i(# bits), 2] rotate: T spec: int-pot inst: name: succ

Additional information

Used simplifier rules: ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j);% ⊦ i * j = j * i; Used terms: n→i(# bits), 2