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Proof step 23 from proof no-dups-keyset-size (previous step)

#(keyset(kar, m0) ++ kar[m0]) = m0 + 1, # keyset(kar, m0) = m0, kar[m1] = kar[m], m0 < # kar, m1 < # kar, m < # kar, m1 ≤ m0, m ≤ m0, ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1] ⊦ m1 = m

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ ¬ ki ∈ s → #(s ++ ki) = # s + 1
Proof step 25:
# keyset(kar, m0) = m0, # keyset(kar, m0) + 1 = m0 + 1, kar[m1] = kar[m], m < # kar, m1 < # kar, m0 < # kar, m ≤ m0, m1 ≤ m0, ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1] ⊦ m1 = m, kar[m0] ∈ keyset(kar, m0)

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: keyset-size-bound0: m < # kar ⊦ # keyset(kar, m) = m + 1 ↔ false;% ⊦ ki ∈ s → s ++ ki = s; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: kar[m0], keyset(kar, m0)