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Proof step 13 from proof ole-last (previous step)

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

Heuristic cut applied rule cut formula

And obtained the following new goals

Proof step 14:
a ∈ y, 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

fma: a ∈ y

Additional information

Used simplifier rules: le-o: ⊦ ordered≤(a ');%le-r: ⊦ ordered≤(a ' + b ' + x) ↔ ¬ b < a ∧ ordered≤(b + x);%ole-two: ⊦ ordered≤(a ' + b ') ↔ ¬ b < a;%ni: ⊦ x + a ' = y + z ∧ ¬ a ∈ z → z = []; (forward) from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%nen: ⊦ ¬ b < a ∧ a = c → ¬ b < c; (forward) from specification oelem%app: ⊦ x + [] = x; from specification list%c-05: ⊦ a ' = x + b ' ↔ a = b ∧ x = []; from specification list%in-empty: ⊦ ¬ a ∈ []; from specification list%nnn: ⊦ ¬ c < b ∧ ¬ a < c → ¬ a < b; (forward) from specification oelem%Nil: ⊦ [] + x = x; from specification list%c: ⊦ a + x = a ' + x; from specification list Used formulas: a ∈ y