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Proof step 7 from proof sep-forall-reduce (previous step)

((nP ⊗_(n, m0 + 1) * P0))(zh), ∀ nP, n, P0, zh, nP2. ((nP ⊗_(n, m0) * P0))(zh) ∧ (∀ n0, P. n0 < m0 ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh)) → ((nP2 ⊗_(n, m0) * P0))(zh), ∀ n0, P. ¬ m0 < n0 ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh) ⊦ ((nP2 ⊗_(n, m0) * nP2(n + m0) * P0))(zh)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1)
Proof step 9:
(((nP ⊗_(n, m0) * nP(n + m0) * nP ⊗_(n + m0 + 1, m0 + 1 - m0 - 1)) * P0))(zh), m0 < m0 + 1, ∀ n0, P. ¬ m0 < n0 ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh), ∀ nP, n, P0, zh, nP2. ((nP ⊗_(n, m0) * P0))(zh) ∧ (∀ n0, P. n0 < m0 ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh)) → ((nP2 ⊗_(n, m0) * P0))(zh) ⊦ ((nP2 ⊗_(n, m0) * nP2(n + m0) * P0))(zh)

Rule argument

spec: iter-n, inst: zheap name: Sep-unfold seq: ⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1) subst: [m0, m, n, nP0] → [m0, m0 + 1, n, nP] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: m0, m0 + 1, n, nP