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Proof step 38 from proof cast-mod-neg-high (previous step)

# int2bits(j) = i→n(i) + 1, 0 < j + 2 ^ i, j < j % 2 ^ i, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < 0, 0 < i, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ bits2int(int2bits(j).rest) = j % 2 ^ i, j = ~1, i = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ j < ~1 → bits2int(int2bits(j).rest) = j + 2 ^ (n→i(# int2bits(j)) - 1)
Proof step 40:
j < ~1, # int2bits(j) = i→n(i) + 1, 0 < j + 2 ^ i, j < j % 2 ^ i, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < 0, 0 < i, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0 ⊦ j + 2 ^ (n→i(# int2bits(j)) - 1) = j % 2 ^ i, j = ~1, i = 1

Rule argument

seq: ⊦ j < ~1 → bits2int(int2bits(j).rest) = j + 2 ^ (n→i(# int2bits(j)) - 1) subst: [j] → [j] rotate: F spec: int2bits inst: name: int2int-rest-01

Additional information

Used simplifier rules: ⊦ ¬ i < j → (i ≠ j ↔ j < i);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: j