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Proof step 5 from proof valid-indices-nodups-h (previous step)

dups(ar.indicesWhere(φ, n0 +1)), n0 +1 ≤ # ar, ∀ ar, φ. ¬(dups(ar.indicesWhere(φ, n0)) ∧ n0 ≤ # ar) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
⊦ n < # ar → ar.indicesWhere(φ, n + 1) = ar.indicesWhere(φ, n) + (φ(ar[n]) ⊃ n + [];[])
Proof step 7:
n0 < # ar, dups(ar.indicesWhere(φ, n0) + (φ(ar[n0]) ⊃ n0 ';[])), ∀ ar0, φ0. ¬ dups(ar0.indicesWhere(φ0, n0)) ∨ ¬ n0 ≤ # ar0 ⊦

Rule argument

Additional information

Used simplifier rules: indiceswhere-rec: ⊦ n < # ar → ar.indicesWhere(φ, n + 1) = ar.indicesWhere(φ, n) + (φ(ar[n]) ⊃ n + [];[]);% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ nx + [] = nx; (s)% ⊦ n + nx = n ' + nx; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m +1 = m + 1; (s)