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Proof step 35 from proof int2bytes-first-zerobyte (previous step)

0 < i, 0 ≤ i, 255 < 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)) % 8 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ 2 ^ i ≤ j ∧ 0 < i → int2bits(j) = int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest
Proof step 37:
2 ^ 8 ≤ i ∧ 0 < 8, 0 < i, 0 ≤ i, 255 < 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 / 2 ^ 8) + fillsgn(i→n(8 + 1), int2bits(i % 2 ^ 8)).rest)) % 8 = 1

Rule argument

seq: ⊦ 2 ^ i ≤ j ∧ 0 < i → int2bits(j) = int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest subst: [i, j] → [8, i] rotate: F spec: int2bits inst: name: int2bits-rec-2pot-pos

Additional information

Used simplifier rules: int2bytes-byte: ~128 ≤ i, i < 128 ⊦ int2bytes((* i)) = i2b(i) ';% ⊦ ¬ i ≤ j ↔ j < i;%0 < j ⊦ 0 ≤ i ∧ i < j → i / (* j) = 0;% ⊦ by '.first = by;%~128 ≤ i, i ≤ 127, ~128 ≤ j, j ≤ 127 ⊦ i2b((* i)) = i2b((* j)) ↔ i = j;%i + 1 = i0 ⊦ j < (* i0) → ¬ (* i) < j; Used terms: 8, i