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Proof step 17 from proof valid-idx-remlast (previous step)

kar[m4] = key-index(key1, m3), ar[m3] = elemindex(a0, m4), kar[0, key-index(key, n)][m4] = key-index(key0, m1), kar[m0] = key-index(key, n), ar[n] = elemindex(a1, m0), ar[n, elemindex(a1, 0)][m1] = elemindex(a, m2), m3 < # ar, m4 + 1 < # kar, m4 < m0, m4 < # kar, m0 < # kar, n < # ar, ∀ m. ¬ m0 < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m1 < # ar ∧ m2 = m4

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 18:
ar[n, elemindex(a1, 0)][m1] = elemindex(a, m2), ar[n] = elemindex(a1, m0), kar[m0] = key-index(key, n), kar[0, key-index(key, n)][m4] = key-index(key0, m1), ar[m3] = elemindex(a0, m4), kar[m4] = key-index(key1, m3), n < # ar, m0 < # kar, m4 < # kar, m4 < m0, m4 + 1 < # kar, m3 < # ar, ∀ m. ¬ m0 < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m4 = 0, m1 < # ar ∧ m2 = m4

Rule argument

fma: m4 = 0

Additional information

Used simplifier rules: ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < # ar → ar[m, ei][m] = ei; (s)% ⊦ elemindex(a, m) = elemindex(a0, m0) ↔ a = a0 ∧ m = m0; (ws) Used formulas: m4 = 0