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Proof step 5 from proof keys-drop-lower (previous step)

m + n + nused0 < # zbrar, ∀ m, n, zbrar, zbr. m + n + nused0 ≤ # zbrar → keys(zbrar[m, zbr], m + n, nused0) = keys(zbrar, m + n, nused0) ∨ n = 0 ⊦ keys(zbrar[m, zbr], m + n, nused0 + 1) = keys(zbrar, m + n, nused0 + 1), n = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ m < # zbrar → keys(zbrar, m, nused + 1) = keys(zbrar, m + 1, nused) ++ zbrar[m].key
Proof step 7:
m + n + nused0 < # zbrar, m + n < # zbrar[m, zbr], ∀ m, n, zbrar, zbr. m + n + nused0 ≤ # zbrar → keys(zbrar[m, zbr], m + n, nused0) = keys(zbrar, m + n, nused0) ∨ n = 0 ⊦ keys(zbrar[m, zbr], (m + n) + 1, nused0) ++ zbrar[m, zbr][m + n].key = keys(zbrar, m + n, nused0 + 1), n = 0

Rule argument

spec: <toplevel> name: keys-rec seq: ⊦ m < # zbrar → keys(zbrar, m, nused + 1) = keys(zbrar, m + 1, nused) ++ zbrar[m].key subst: [nused, m, zbrar] → [nused0, m + n, zbrar[m, zbr]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # zbrar[n, zbr] = # zbrar; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused0, m + n, zbrar[m, zbr]