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Proof step 1 from proof Inv-end-reduce (conclusion)

⊦ Inv-end(A, A0, i, j, y) → ( Inv-end(A1, A2, i0, j0, y0) ↔ # A ≠ 0 ∧ # A = # A0 ∧ j ≤ i ∧ (j ≠ i → y ≤ A[j + 1]) ∧ sorted(A, 0, (j = i ⊃ i;i + 1)) ∧ perm(A[j, y], A0) → # A1 ≠ 0 ∧ # A1 = # A2 ∧ j0 ≤ i0 ∧ (j0 ≠ i0 → y0 ≤ A1[j0 + 1]) ∧ sorted(A1, 0, (j0 = i0 ⊃ i0;i0 + 1)) ∧ perm(A1[j0, y0], A2))

System applied rule normalize

And obtained the following new goals

Proof step 2:
Inv-end(A, A0, i, j, y) ⊦ Inv-end(A1, A2, i0, j0, y0) ↔ # A ≠ 0 ∧ # A = # A0 ∧ j ≤ i ∧ (j ≠ i → y ≤ A[j + 1]) ∧ sorted(A, 0, (j = i ⊃ i;i + 1)) ∧ perm(A[j, y], A0) → # A1 ≠ 0 ∧ # A1 = # A2 ∧ j0 ≤ i0 ∧ (j0 ≠ i0 → y0 ≤ A1[j0 + 1]) ∧ sorted(A1, 0, (j0 = i0 ⊃ i0;i0 + 1)) ∧ perm(A1[j0, y0], A2)

Proof step 1 from proof Inv-end-reduce (conclusion)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information