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Proof step 78 from proof cast-mod-neg-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 79:
⊦ i + 1 ≠ 0 → (~1 ≤ i ↔ 0 ≤ i)
Proof step 80:
j % 2 ^ i + 1 ≠ 0, i→n(i) = # int2bits(j % 2 ^ i).rest, i→n(i) < # int2bits(j % 2 ^ i), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0, j + 2 ^ i ≤ 0, j + 2 ^ i < i, 0 ≤ j % 2 ^ i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(int2bits(j % 2 ^ i).rest) = j % 2 ^ i + 2 ^ i

Rule argument

seq: ⊦ i + 1 ≠ 0 → (~1 ≤ i ↔ 0 ≤ i) subst: [i] → [j % 2 ^ i] rotate: F spec: int-abs inst: name: mone-07

Additional information

Used simplifier rules: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ m < n ∧ m = n0 → n0 < n; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ # bits = 0 ↔ bits = [];% ⊦ int2bits(j) ≠ [];%i < 0 ⊦ (* i) + j = j - ~ i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ 2 ^ i < 1 ↔ i < 0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: j % 2 ^ i