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Proof step 3 from proof sorted-eq-entry (previous step)

zbrar[m0] = mkzentry(key, adr0), zbrar[m1] = mkzentry(key, adr1), m1 < m, m0 < m, m1 < # zbrar, m0 < # zbrar, sorted(zbrar, 0, m), m ≤ # zbrar ⊦ m0 = m1 ∧ adr0 = adr1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
n + m ≤ # zbrar ⊦ sorted(zbrar, n, m) ↔ (∀ m0, m1. m0 < m1 ∧ m1 < m → zbrar[n + m0].key < zbrar[n + m1].key)
Proof step 5:
zbrar[m1] = mkzentry(key, adr1), zbrar[m0] = mkzentry(key, adr0), m0 < # zbrar, m1 < # zbrar, m0 < m, m1 < m, m ≤ # zbrar, 0 + m ≤ # zbrar, ∀ m0, m1. m0 < m1 ∧ m1 < m → zbrar[0 + m0].key < zbrar[0 + m1].key ⊦ m0 = m1 ∧ adr0 = adr1

Rule argument

spec: <toplevel> name: sorted-all seq: n + m ≤ # zbrar ⊦ sorted(zbrar, n, m) ↔ (∀ m0, m1. m0 < m1 ∧ m1 < m → zbrar[n + m0].key < zbrar[n + m1].key) subst: [m, n, zbrar] → [m, 0, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n + 0 = n; (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)% ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: m, 0, zbrar