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Proof step 71 from proof uint2bits-restsgn (previous step)

2 ^ i0 = i + 1, i < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), i * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 < i * 2 ^ n→i(# bits0), 1 < i0 + n→i(# bits0), 0 ≤ i * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * i, i * 2 ^ n→i(# bits0) * 2 + 1 < (i + 1) * 2 ^ n→i(# bits0), (i + 1) * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * i < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ i0, 1 < (i + 1) * 2 ^ n→i(# bits0), 0 < i0 + n→i(# bits0), 0 ≤ i0 + n→i(# bits0), i ≤ 1, 0 < (i + 1) * 2 ^ n→i(# bits0), 0 ≤ (i + 1) * 2 ^ n→i(# bits0), i < i0 + n→i(# bits0), i < (i + 1) * 2 ^ n→i(# bits0), i + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), ∀ j0, i1. i→n(j0) < i→n(2 * (i * 2 ^ n→i(# bits0) * 2 + 1) + 0) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ 4 * (i + 1) * 2 ^ n→i(# bits0) = 1 + 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

User applied rule cut formula

And closed the goal!

Rule argument

fma: i = 1

Additional information

Used simplifier rules: ⊦ ¬ 2 ^ i < 0;% ⊦ i ≤ 0 → (0 < i * j ↔ ¬ ¬(j < 0 ∧ i ≠ 0));% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ 2 ^ i = 2 ↔ i = 1;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 1 * i = i;% ⊦ j0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 ≤ j0 → (i * j + j0 < i0 * j ↔ i < i0);% ⊦ ¬ i < i; Used formulas: i = 1