[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 112 from proof prepost-remove (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key, m), 2exp-size(# kar), m2 < # kar, m1 < # kar, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar, m2 + 1), m1 ≤ m2, 1 ≤ # kar, key-index(key, m) = kar[m2] ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ kar[m2]) = 0, keyset(kar, m2) = (keyset(kar, m2) ++ kar[m2]) -- key-index(key, m), m2 ≠ 0 ∧ m2 ≠ m1

User applied rule case distinction

And closed the goal!

Rule argument

pos: right 3

Additional information

Used simplifier rules: ⊦ m ≤ m; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ (s ++ ki) -- ki = s -- ki; (s)% ⊦ s -- ki = s ↔ ¬ ki ∈ s; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ n < n; (s)%m ≤ m1 ⊦ valid-idx(kar, ar, m + 1) ∧ ar[m0] = elemindex(a, m1) → ¬ key-index(key, m0) ∈ keyset(kar, m); (s)% ⊦ keyset(kar, 0) = ∅; (s)% ⊦ s -- ki = ∅ ↔ s = ∅ ∨ s = ∅ ++ ki; (ws)% ⊦ #(∅ ++ ki) = 1; (s)% ⊦ m ≤ 0 ↔ m = 0; (s)% ⊦ heap-order(kar, 1); (s)% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ ¬ ki ∈ ∅; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic