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Proof step 8 from proof inv-propagate (previous step)

∀ a, b, Pelem. rtc(n0, a, b) ∧ Pelem(a) ∧ (∀ a, b. Pelem(a) ∧ p(a, b) → Pelem(b)) → Pelem(b), rtc(n0 +1, a, b), Pelem(a), ∀ a, b. Pelem(a) ∧ p(a, b) → Pelem(b) ⊦ Pelem(b)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
Pelem(a), p(a, c), rtc(n0, c, b), ∀ a, b, Pelem. rtc(n0, a, b) ∧ Pelem(a) ∧ (∀ a, b. Pelem(a) ∧ p(a, b) → Pelem(b)) → Pelem(b), ∀ a, b. Pelem(a) ∧ p(a, b) → Pelem(b) ⊦ Pelem(b)

Rule argument

Additional information

Used simplifier rules: rtc-rec: ⊦ rtc(n + 1, a, b) ↔ (∃ c. p(a, c) ∧ rtc(n, c, b));%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%succ: ⊦ m +1 = m + 1; from specification nat-basic2