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Proof step 5 from proof Inv-last-inner-init-b (previous step)

# A = # A0, m < # A0, m < i, i + 1 < # A0, perm(A[m + 1, y][2 + m, x], A0), sorted(A, 0, m + 1), sorted(A, 3 + m, i + 2), i ≤ # A0, m + 1 ≠ i → x < A[3 + m], m + 1 ≠ i → ¬ A[3 + m] < A[m] ⊦ x < A[m], x < y, sorted(A[2 + m, x], 2 + m, i + 2) ∧ perm(A[2 + m, x][m + 1, y], A0)

User applied rule case distinction

And obtained the following new goals

Proof step 6:
# A = # A0, i + 1 < # A0, m < i, m < # A0, perm(A[m + 1, y][2 + m, x], A0), sorted(A, 3 + m, i + 2), sorted(A, 0, m + 1), i ≤ # A0, m + 1 ≠ i → ¬ A[3 + m] < A[m], m + 1 ≠ i → x < A[3 + m] ⊦ x < y, x < A[m], sorted(A[2 + m, x], 2 + m, i + 2)
Proof step 8:
# A = # A0, i + 1 < # A0, m < i, m < # A0, perm(A[m + 1, y][2 + m, x], A0), sorted(A, 3 + m, i + 2), sorted(A, 0, m + 1), i ≤ # A0, m + 1 ≠ i → ¬ A[3 + m] < A[m], m + 1 ≠ i → x < A[3 + m] ⊦ x < y, x < A[m], perm(A[2 + m, x][m + 1, y], A0)

Rule argument

pos: right 3

Proof step 5 from proof Inv-last-inner-init-b (previous step)

User applied rule case distinction

And obtained the following new goals

Rule argument

Additional information