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Proof step 9 from proof keyset-size-bound (previous step)

m0 + 1 < #(keyset(kar, m0) ++ kar[m0]), m0 + 1 < # kar, m0 < # kar, # keyset(kar, m0) < # kar, # keyset(kar, m0) ≤ m0 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ¬ ki ∈ s → #(s ++ ki) = # s + 1

Rule argument

spec: set-basic, inst: keyindex-set name: Size-insert seq: ⊦ ¬ ki ∈ s → #(s ++ ki) = # s + 1 subst: [ki, s] → [kar[m0], keyset(kar, m0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m ≤ m0 ∧ n ≤ n0 → ¬ m0 + n0 < m + n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ki ∈ s → s ++ ki = s; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set]% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ ¬ m + n < m; (s) Used terms: kar[m0], keyset(kar, m0)