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Proof step 38 from proof int2bytes-first-zerobyte (previous step)

0 < i, 0 ≤ i, 256 ≤ i, ∀ i0. i→n(i0) < i→n(i) → 0 ≤ i0 → (int2bytes(i0).first = i2b(0) ↔ n→i(# int2bits(i0)) % 8 = 1) ⊦ int2bytes(i / 256).first = i2b(0) ↔ (n→i(# int2bits(i / 256)) + n→i(# fillsgn(9, int2bits(i % 256)).rest)) % 8 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ i0 % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) % j = i % j
Proof step 40:
¬(n→i(# fillsgn(9, int2bits(i % 256)).rest) % 8 = 0 ∧ 0 ≤ n→i(# int2bits(i / 256)) ∧ 0 ≤ n→i(# fillsgn(9, int2bits(i % 256)).rest) ∧ 0 < 8), 0 < i, 0 ≤ i, 256 ≤ i, ∀ i0. i→n(i0) < i→n(i) → 0 ≤ i0 → (int2bytes(i0).first = i2b(0) ↔ n→i(# int2bits(i0)) % 8 = 1) ⊦ int2bytes(i / 256).first = i2b(0) ↔ (n→i(# int2bits(i / 256)) + n→i(# fillsgn(9, int2bits(i % 256)).rest)) % 8 = 1
Proof step 49:
n→i(# fillsgn(9, int2bits(i % 256)).rest) % 8 = 0 ∧ 0 ≤ n→i(# int2bits(i / 256)) ∧ 0 ≤ n→i(# fillsgn(9, int2bits(i % 256)).rest) ∧ 0 < 8, 0 < i, 0 ≤ i, 256 ≤ i, ∀ i0. i→n(i0) < i→n(i) → 0 ≤ i0 → (int2bytes(i0).first = i2b(0) ↔ n→i(# int2bits(i0)) % 8 = 1) ⊦ int2bytes(i / 256).first = i2b(0) ↔ n→i(# int2bits(i / 256)) % 8 = 1

Rule argument

seq: ⊦ i0 % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) % j = i % j subst: [j, i0, i] → [8, n→i(# fillsgn(9, int2bits(i % 256)).rest), n→i(# int2bits(i / 256))] rotate: F spec: int-div inst: name: mod-add-pos

Additional information

Used terms: 8, n→i(# fillsgn(9, int2bits(i % 256)).rest), n→i(# int2bits(i / 256))