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Proof step 7 from proof mod-two-ge-01 (previous step)

j < i, 0 < i, 0 ≤ j ⊦ 2 ^ j * 2 ^ (i - j) = 2 ^ j * ((2 ^ j * 2 ^ (i - j) - 1) / 2 ^ j + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < i0 ∧ i0 < j ∧ 0 < i → (j * i - i0) / j = i - 1
Proof step 9:
0 < i, j < i, 0 ≤ j ⊦ 2 ^ j * 2 ^ (i - j) = 2 ^ j * ((2 ^ j * 2 ^ (i - j) - 1) / 2 ^ j + 1), 0 < 1 ∧ 1 < 2 ^ j ∧ 0 < 2 ^ (i - j)

Rule argument

spec: int-div name: mult-05 seq: ⊦ 0 < i0 ∧ i0 < j ∧ 0 < i → (j * i - i0) / j = i - 1 subst: [i0, i, j] → [1, 2 ^ (i - j), 2 ^ j] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i + (j - i) = j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 1, 2 ^ (i - j), 2 ^ j