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Proof step 7 from proof add-eq2pot-01 (previous step)

0 ≤ i, 2 ^ i2 = i1 + 1, i + 1 = i2 + i0, 0 ≤ i0, 0 ≤ i, 0 ≤ j ⊦ 2 ^ (i + 1) = 2 ^ i0 + j ↔ j = 2 ^ i0 * i1 ∧ 0 ≤ i2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 ≤ j ∧ 0 ≤ i ∧ 0 ≤ i0 → (2 ^ i = 2 ^ i0 + j ↔ j = 2 ^ i0 * (2 ^ (i - i0) - 1) ∧ i0 ≤ i)

Rule argument

spec: int-pot inst: name: add-eq2pot seq: ⊦ 0 ≤ j ∧ 0 ≤ i ∧ 0 ≤ i0 → ( 2 ^ i = 2 ^ i0 + j ↔ j = 2 ^ i0 * (2 ^ (i - i0) - 1) ∧ i0 ≤ i) subst: [j, i0, i] → [j, i0, i + 1] flags: rewrite, current

Additional information

Used simplifier rules: succ: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-add: ⊦ i + j - i = j; from specification int-abs%lep: ⊦ i ≤ i + j ↔ 0 ≤ j; from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: j, i0, i + 1