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Proof step 10 from proof geom-mod (previous step)

(1 - i ^ i0) % (1 - i) = 0, 0 ≤ i0, ∀ j0, i. i→n(j0) < i→n(i0 + 1) → 0 ≤ j0 → (1 - i ^ j0) % (1 - i) = 0 ∨ i = 1 ⊦ i = 1, ~1 = i0, (1 - i ^ i0 * i) % (1 - i) = 0

User applied rule cut formula

And obtained the following new goals

Proof step 11:
1 - i ^ i0 * i = (1 - i ^ i0) * i - (i - 1), (1 - i ^ i0) % (1 - i) = 0, 0 ≤ i0, ∀ j0, i. i→n(j0) < i→n(i0 + 1) → 0 ≤ j0 → (1 - i ^ j0) % (1 - i) = 0 ∨ i = 1 ⊦ i = 1, ~1 = i0, (1 - i ^ i0 * i) % (1 - i) = 0

Rule argument

fma: 1 - i ^ i0 * i = (1 - i ^ i0) * i - (i - 1)

Additional information

Used simplifier rules: sub-sub-04: ⊦ i - (j - j0) = i + j0 - j; from specification int-abs%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult%sub-add: ⊦ i + j - i = 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%one: ⊦ 1 * i = i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: 1 - i ^ i0 * i = (1 - i ^ i0) * i - (i - 1)