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Proof step 31 from proof not-always-decrease (previous step)

N = N'' + 1 until A ≪ a, n = N + 1, boolvar → a0 = a2 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → N = N'' +1 until A ≪ a → (∃ a1. ◇ (a1 = A ∧ A ≪ a)) ⊦ ∃ a1. a1 = a0 ∧ a0 ≪ a ∨ ◇ (a1 = A ∧ A ≪ a)

User applied rule apply induction

And obtained the following new goals

Proof step 32:
∃ a1. ◇ (a1 = A ∧ A ≪ a), N = N'' + 1 until A ≪ a, n = N + 1, boolvar → a0 = a2 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → N = N'' +1 until A ≪ a → (∃ a1. ◇ (a1 = A ∧ A ≪ a)) ⊦ ∃ a1. a1 = a0 ∧ a0 ≪ a ∨ ◇ (a1 = A ∧ A ≪ a)

Rule argument

subst: [N, A, a] → [N, A, a]

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ 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: N, A, a