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Proof step 12 from proof ole-last (previous step)

x0 + a ' = a1 + y ∧ x0 + a ' ≠ [], ordered≤(x0 + a '), ∀ a, b. ordered≤(x0 + a ') → (ordered≤(x0 + a ' + b ') ↔ ¬ b < a) ⊦ b < a, ordered≤((a0 ' + x0) + a ' + b '), a1 < a0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
x0 + a ' = a1 ' + y, ordered≤(a1 ' + y), ∀ a, b. ordered≤(x0 + a ') → (ordered≤(x0 + a ' + b ') ↔ ¬ b < a), a = a1 ∨ a ∈ y, a1 ∈ x0 ∨ a1 = a ⊦ a1 < a0, ordered≤((a0 ' + x0) + a ' + b '), b < a

Rule argument

Additional information

Used simplifier rules: cons-02: ⊦ ordered≤(a ' + x) → ordered≤(x);%app-01: ⊦ ordered≤(x + y) → ordered≤(y);%in-17: ⊦ a ∈ b ' ↔ a = b; from specification list%in-append: ⊦ a ∈ x + y ↔ ¬ ¬(a ∈ x ∨ a ∈ y); from specification list%in-rec: ⊦ a ∈ b ' + x ↔ ¬ ¬(a = b ∨ a ∈ x); from specification list%in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list%in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%e-02: ⊦ x + a ' ≠ []; from specification list%c: ⊦ a + x = a ' + x; from specification list