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Proof step 13 from proof cast-size-fits (previous step)

2 ^ i0 + j < 0, j ≤ 0, j < 2 ^ i0, # int2bits(j) ≤ i→n(i0) + 1, 0 ≤ i0 ⊦ cast(i0 + 1, j) = j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ i < ~1 → (# int2bits(i) ≤ m ↔ 0 ≤ i + 2 ^ (n→i(m) - 1))

Rule argument

seq: ⊦ i < ~1 → (# int2bits(i) ≤ m ↔ 0 ≤ i + 2 ^ (n→i(m) - 1)) subst: [m, i] → [i→n(i0) + 1, j] rotate: F spec: int2bits inst: name: int2bits-lelen-03

Additional information

Used simplifier rules: ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i + 0 = i;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ 0 ≤ i → n→i(i→n(i)) = i;% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i + j < j ↔ i < 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j;% ⊦ i < 0 ∧ 0 ≤ j → i < j;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;%i < 0 ⊦ (* i) + j = j - ~ i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ 2 ^ i < 1 ↔ i < 0;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: i→n(i0) + 1, j