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Proof step 42 from proof uint2bits-add-low (previous step)

j ≤ 2 ^ (i - 1), j ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < j + 2 ^ i, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), j < 0, 0 < j + 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 2 ^ (i - 2) < (1 - j) / 2, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = false ' + fillsgn(i→n(i), int2bits(j)), i = 1, j = 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 43:
1 - j = i1 * 2 + j0 ∧ 0 ≤ j0 ∧ j0 < 2 ∧ 0 < 2 ∧ 0 ≤ 1 - j, j ≤ 2 ^ (i - 1), j ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < j + 2 ^ i, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), j < 0, 0 < j + 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 2 ^ (i - 2) < i1, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = false ' + fillsgn(i→n(i), int2bits(j)), i = 1, j = 0

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [1 - j, 2, i1, j0] rotate: T spec: int-div inst: name: div-elim-pos

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i + 0 = i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j ∧ ¬ i0 < j → ¬ i0 < i;%⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j); (elim) Used terms: 1 - j, 2, i1, j0