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Proof step 9 from proof exinpot (previous step)

0 < i, i ^ i0 < j, 1 < i, 1 < j, 0 < j, 0 < i ^ i0, 1 < i ^ i0 * i, 0 ≤ i0, j ≤ i ^ i0 * i, ∀ j0, i. i→n(j0) < i→n(j) → 1 < i ∧ 0 < j0 → (∃ i0. 0 ≤ i0 ∧ i ^ i0 ≤ j0 ∧ j0 < i ^ (i0 + 1)) ⊦ ∃ i0. 0 ≤ i0 ∧ i ^ i0 ≤ j ∧ j < i ^ i0 * i, 0 ≤ i0 ∧ i ^ i0 ≤ j ∧ j < i ^ i0 * i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
1 < i ^ i0 * i, 0 < i ^ i0, 0 < i, 1 < i, 0 ≤ i0, ∀ j0, i1. i→n(j0) < i→n(i ^ i0 * i) → 1 < i1 ∧ 0 < j0 → (∃ i0. 0 ≤ i0 ∧ i1 ^ i0 ≤ j0 ∧ j0 < i1 ^ (i0 + 1)) ⊦ ∃ i1. 0 ≤ i1 ∧ i ^ i1 ≤ i ^ i0 * i ∧ i0 < i1

Rule argument

Additional information

Used simplifier rules: ls-pos: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0);% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ 0 < i → (i < i * j ↔ 1 < j); (s)%i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j; (s)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i ≤ i; (s)% ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 1 ≤ i ↔ 0 < i; (s)% ⊦ 0 < i → (i ≤ i * j ↔ 1 ≤ j); (s)%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i < j → ¬ j < i; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)