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Proof step 8 from proof ordered-ex-divide-lsle (previous step)

ordered≤(a0 ' + y + z), ∀ b. b ∈ y → b < a, ∀ b. b ∈ z → b = a ∨ a < b ⊦ ∃ y0, z0. a0 ' + y + z = y0 + z0 ∧ (∀ b. b ∈ y0 → b < a) ∧ (∀ b. b ∈ z0 → b = a ∨ a < b), a0 ' + y + z = (a0 ' + y) + z ∧ (∀ b. b ∈ a0 ' + y → b < a) ∧ (∀ b. b ∈ z → b = a ∨ a < b)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
ordered≤(a0 ' + y + z), ∀ b. b ∈ z → b = a ∨ a < b, ∀ b. b ∈ y → b < a, b = a0 ∨ b ∈ y ⊦ b < a, ∃ y0, z0. a0 ' + y + z = y0 + z0 ∧ (∀ b. b ∈ y0 → b < a) ∧ (∀ b. b ∈ z0 → b = a ∨ a < b)

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ b ' ↔ a = b; (s)% ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list