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Proof step 27 from proof payload-mod (previous step)

zbrar[m] = mkzentry(zbrar[(m + 1) + n0].key, adr0), payload(zbrar[(m + 1) + n0, mkzentry(zbrar[(m + 1) + n0].key, adr)], m + 1, nused0) = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr], m < # zbrar, m + nused0 < # zbrar, n0 < nused0, sorted(zbrar, m, nused0 + 1) ⊦ payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr0] = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
n + m ≤ # zbrar ⊦ sorted(zbrar, n, m) ↔ (∀ m0, m1. m0 < m1 ∧ m1 < m → zbrar[n + m0].key < zbrar[n + m1].key)
Proof step 29:
zbrar[m] = mkzentry(zbrar[(m + 1) + n0].key, adr0), payload(zbrar[(m + 1) + n0, mkzentry(zbrar[(m + 1) + n0].key, adr)], m + 1, nused0) = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr], n0 < nused0, m + nused0 < # zbrar, m < # zbrar, m + nused0 + 1 ≤ # zbrar, ∀ m0, m1. m0 < m1 ∧ m1 < nused0 + 1 → zbrar[m + m0].key < zbrar[m + m1].key ⊦ payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr0] = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr]

Rule argument

spec: zbranch-array-sorted 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] → [nused0 + 1, m, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (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: nused0 + 1, m, zbrar