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Proof step 10 from proof exin2pot-neg (previous step)

j = ~(2 ^ i), j < 0, 0 < 2 ^ i * 2 + j, 0 ≤ i, 0 ≤ 2 ^ i * 2 + j ⊦ ∃ i. ~1 ≤ i ∧ j + 2 ^ i < 0 ∧ 0 ≤ 2 ^ (i + 1) + j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
0 ≤ 2 ^ i, 0 ≤ i ⊦ ∃ i0. ~1 ≤ i0 ∧ 2 ^ i0 < 2 ^ i ∧ i ≤ i0 + 1

Rule argument

Additional information

Used simplifier rules: pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i;%two-le: ⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j);%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s)% ⊦ i + 0 = i; (s)% ⊦ i - j < i0 ↔ i < j + i0; (s)% ⊦ ~ i < 0 ↔ 0 < i; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 1 * i = i; (s)% ⊦ i * j - i = i * (j - 1); (s)% ⊦ i + ~ j = i - j; (s)