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Proof step 106 from proof cast-mod-neg-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 107:
0 ≤ i, ~(2 ^ (i + 1)) ≤ j, j < ~(2 ^ i) ⊦ # int2bits(j) = i→n(i + 2)
Proof step 108:
¬(0 ≤ i - 1 ∧ ~(2 ^ ((i - 1) + 1)) ≤ j % 2 ^ i ∧ j % 2 ^ i < ~(2 ^ (i - 1))), j % 2 ^ i < ~1, i→n(i) = # int2bits(j % 2 ^ i).rest, i→n(i) < # int2bits(j % 2 ^ i), j + 2 ^ i < i, j + 2 ^ i ≤ 0, 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 ^ i + 2 ^ (n→i(# int2bits(j % 2 ^ i)) - 1) = j % 2 ^ i, j = ~1, i = 1, j % 2 ^ i = 0

Rule argument

seq: 0 ≤ i, ~(2 ^ (i + 1)) ≤ j, j < ~(2 ^ i) ⊦ # int2bits(j) = i→n(i + 2) subst: [i, j] → [i - 1, j % 2 ^ i] rotate: F spec: int2bits inst: name: int2bits-len-neg

Additional information

Used simplifier rules: ⊦ i + (j - i0) = i + j - i0;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ 0 < i → n→i(i→n(i)) = i;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ i + 0 = i;% ⊦ i = i + j ↔ j = 0;% ⊦ 0 < j → (i ^ j = 0 ↔ i = 0);% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i + (j - i) = j;% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ i < ~ j ↔ i + j < 0;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: i - 1, j % 2 ^ i