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Proof step 42 from proof cast-add-2pot-h (previous step)

j < 0, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i + j, j + 2 ^ (i - 1) < 0, 2 ^ (i - 1) ≤ (2 ^ i + j) % 2 ^ i ⊦ cast(i, j + 2 ^ i) = j + 2 ^ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ 0 ≤ i ∧ i0 < 0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j + j
Proof step 44:
¬(2 ^ i % 2 ^ i = 0 ∧ 0 ≤ 2 ^ i + j ∧ 0 ≤ 2 ^ i ∧ j < 0 ∧ j % 2 ^ i ≠ 0 ∧ 0 < 2 ^ i), j < 0, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i + j, j + 2 ^ (i - 1) < 0, 2 ^ (i - 1) ≤ (2 ^ i + j) % 2 ^ i ⊦ cast(i, j + 2 ^ i) = j + 2 ^ i
Proof step 47:
2 ^ i % 2 ^ i = 0 ∧ 0 ≤ 2 ^ i + j ∧ 0 ≤ 2 ^ i ∧ j < 0 ∧ j % 2 ^ i ≠ 0 ∧ 0 < 2 ^ i, j < 0, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i + j, j + 2 ^ (i - 1) < 0, 2 ^ (i - 1) ≤ j % 2 ^ i + 2 ^ i ⊦ cast(i, j + 2 ^ i) = j + 2 ^ i

Rule argument

seq: ⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ 0 ≤ i ∧ i0 < 0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j + j subst: [j, i0, i] → [2 ^ i, j, 2 ^ i] rotate: F spec: int-div inst: name: mod-add1-posneg-pos

Additional information

Used terms: 2 ^ i, j, 2 ^ i