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Proof step 8 from proof not-always-decrease (previous step)

∀ a1. a1 ≪ a0 → \A ∀ A, a. a1 ≠ A ∨ ¬ □ ⟨a := A⟩ (◇ A ≪ a) ∨ a ≠ A, □ ⟨a0 := A⟩ (◇ A ≪ a0), a0 = A ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 9:
∃ a1. ◇ (a1 = A ∧ a1 ≪ a0), ∀ a1. a1 ≪ a0 → \A ∀ A, a. a1 ≠ A ∨ ¬ □ ⟨a := A⟩ (◇ A ≪ a) ∨ a ≠ A, □ ⟨a0 := A⟩ (◇ A ≪ a0), a0 = A ⊦
Proof step 18:
∀ a1. a1 ≪ a0 → \A ∀ A, a. a1 ≠ A ∨ ¬ □ ⟨a := A⟩ (◇ A ≪ a) ∨ a ≠ A, □ ⟨a0 := A⟩ (◇ A ≪ a0), a0 = A ⊦ ∃ a1. ◇ (a1 = A ∧ a1 ≪ a0)

Rule argument

fma: ∃ a1. ◇ (a1 = A ∧ a1 ≪ a0)

Additional information

Used formulas: ∃ a1. ◇ (a1 = A ∧ a1 ≪ a0)