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Proof step 12 from proof ordered-ex-divide-lsle (previous step)

ordered≤(a0 ' + y + z), ∀ b. b ∈ y → b < a, ∀ b. b ∈ z → b = a ∨ a < b ⊦ a0 < a, a0 ' + y + z = [] + a0 ' + y + z ∧ (∀ b. b ∈ [] → b < a) ∧ (∀ b. b ∈ a0 ' + y + z → b = a ∨ a < b)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
b < a, ordered≤(a0 ' + y + z), ∀ b. b ∈ z → b = a ∨ a < b, ∀ b. b ∈ y → b < a, b ∈ y ∨ b ∈ z ⊦ b = a, a0 < a

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a < b ↔ ¬ b < a); (ss)% ⊦ a ∈ b ' ↔ a = b; (s)% ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)% ⊦ ¬ a ∈ []; (s)% ⊦ x + y = x + z ↔ y = z; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ [] + x = x; (s)