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Proof step 30 from proof max-com (previous step)

max(x3 + a0 ' + x) = (max(x3 + x) < a0 ⊃ a0;max(x3 + x)), max(x + a ' + x3) = (max(x3 + x) < a ⊃ a;max(x3 + x)), max(x + x3) = max(x3 + x), ∀ x1, x2. # x1 + # x2 < #(a + x3) + #(a0 + x) → true → x1 + x2 = [] ∨ max(x1 + x2) = max(x2 + x1) ⊦ ((max(x3 + x) < a0 ⊃ a0;max(x3 + x)) < a ⊃ a;(max(x3 + x) < a0 ⊃ a0;max(x3 + x))) = ((max(x3 + x) < a ⊃ a;max(x3 + x)) < a0 ⊃ a0;(max(x3 + x) < a ⊃ a;max(x3 + x))), x3 + x = [], x + x3 = []

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 31:
¬ (max(x3 + x) < a0 ⊃ a0;max(x3 + x)) < a, max(x3 + a0 ' + x) = (max(x3 + x) < a0 ⊃ a0;max(x3 + x)), max(x + a ' + x3) = (max(x3 + x) < a ⊃ a;max(x3 + x)), max(x + x3) = max(x3 + x), ∀ x1, x2. # x1 + # x2 < #(a + x3) + #(a0 + x) → true → x1 + x2 = [] ∨ max(x1 + x2) = max(x2 + x1) ⊦ (max(x3 + x) < a0 ⊃ a0;max(x3 + x)) = ((max(x3 + x) < a ⊃ a;max(x3 + x)) < a0 ⊃ a0;(max(x3 + x) < a ⊃ a;max(x3 + x))), x3 + x = [], x + x3 = []

Rule argument

pos: right 1

Additional information

Used simplifier rules: lel: ⊦ a < b ∧ a = c → c < b; (forward) from specification oelem%transitivity: ⊦ a < b ∧ b < c → a < c; (forward) from specification oelem%ls-le: ⊦ a < b → (¬ b < a ↔ true); from specification oelem%nen: ⊦ ¬ b < a ∧ a = c → ¬ b < c; (forward) from specification oelem%nnn: ⊦ ¬ c < b ∧ ¬ a < c → ¬ a < b; (forward) from specification oelem%irreflexivity: ⊦ ¬ a < a; from specification oelem