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Proof step 5 from proof cast-min (previous step)

0 < i, 2 ^ (i - 1) + cast(i, j) < 0, ~1 ≤ cast(i, j) ⊦

User applied rule insert lemma

And obtained the following new goals

Proof step 6:
⊦ 0 < i → cast(i, j) < 2 ^ (i - 1)

Rule argument

name: cast-max subst: [i, j] → [i, j]

Additional information

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