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Proof step 142 from proof uint2bits-restsgn (previous step)

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

User applied rule apply induction

And obtained the following new goals

Proof step 143:
0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i ≤ i0 * 2, 0 < i, 0 ≤ 2 ^ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i0, i0 < 2 * 2 ^ i, false ∈ uint2bits(2 * 2 ^ i - i0 + 1).rest, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ i→n(i0 + 1) < i→n(i0 * 2 + 1), 4 * 2 ^ i = 2 * (i0 + 1), sgn(uint2bits(2 * 2 ^ i - i0 + 1).rest) = uint2bits(2 * 2 ^ i - i0 + 1).rest
Proof step 146:
0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i ≤ i0 * 2, 0 < i, 0 ≤ 2 ^ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i0, i0 < 2 * 2 ^ i, false ∈ uint2bits(2 * 2 ^ i - i0 + 1).rest, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 0 < i - 1 ∧ 0 ≤ i0 + 1 ∧ 2 ^ (i - 1) < i0 + 1 ∧ i0 + 1 < 2 * 2 ^ (i - 1), 4 * 2 ^ i = 2 * (i0 + 1), sgn(uint2bits(2 * 2 ^ i - i0 + 1).rest) = uint2bits(2 * 2 ^ i - i0 + 1).rest

Rule argument

subst: [j0, i] → [i0 + 1, i - 1]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j = j + i;% ⊦ i + (j - i0) = i + j - i0;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j; Used terms: i0 + 1, i - 1