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Proof step 142 from proof uint2bits-add-high (previous step)

0 ≤ 2 ^ (i2 - 1), 0 < 2 ^ (i2 - 1), 0 < i1 * 2 + i2, 0 ≤ i1 * 2 + i2, 0 ≤ i1 + i2, 2 < 2 ^ i2, 1 < 2 ^ (i2 - 1), 0 < i1, 2 ^ i2 + 2 < 2 * (2 ^ i2 + i1), 1 < 2 ^ i2 + i1, 2 ^ i2 + 1 < 3 * i1, 2 ^ i2 ≤ i1 * 2, 2 ^ i2 ≤ 2 * (2 ^ i2 + i1), 2 ^ i2 < i1 * 2 + i2, i1 ≤ 2 ^ i2, i1 < 2 ^ i2, 2 ^ i2 < i1 * 2, 2 ^ i2 < 3 * i1, i1 < 2 ^ i2 * 2, 1 < i1 + 2 ^ i2 * 2, i1 + 1 < 2 ^ i2 * 2, 0 < i2, i1 ≤ 2 ^ (i2 - 1), 0 ≤ 2 ^ i2 + i1, 0 ≤ 2 ^ i2, i1 ≤ 2 * (2 ^ i2 + i1), i1 < 2 * (2 ^ i2 + i1), 1 < i1 + i2, 0 ≤ i1 + 2 ^ i2 * 2, 1 < 2 ^ i2, 0 < 2 ^ i2, 3 < i1 * 2 + i2, 1 < i1, ∀ i0, j. i→n(i0) < i→n(i2 + 1) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ uint2bits(2 ^ i2 - i1) + false ' = sgn(int2bits(~(i1 * 2)).rest), 0 = i2, i1 = 0, 2 ^ i2 * 2 = 1 + i1 * 2

User applied rule insert elim lemma

And closed the goal!

Rule argument

seq: ⊦ i0 = i - j ↔ i = i0 + j subst: [i, j, i0] → [i2, 1, i] rotate: T spec: int-abs inst: name: elim-sub

Additional information

Used simplifier rules: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;%i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ 0 < i → (0 < i * j ↔ 0 < j);%i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j;% ⊦ i * j + i = i * (j + 1);% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);% ⊦ i + j < (* i1) * i + j0 ↔ j < (i1 - 1) * i + j0;% ⊦ 1 * i = i;% ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0);% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ j ≤ (* i1) * j + i ↔ 0 ≤ (i1 - 1) * j + i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: i2, 1, i