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Proof step 22 from proof max-com (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ x ≠ [] → max(a ' + x) = (max(x) < a ⊃ a;max(x))
Proof step 24:
max(x + a ' + x3) = (max(x3 + x) < a ⊃ a;max(x3 + x)), max(x3 + a0 ' + x) = (max(x + x3) < a0 ⊃ a0;max(x + x3)), ∀ x1, x2. # x1 + # x2 < #(a + x3) + #(a0 + x) → true → x1 + x2 = [] ∨ max(x1 + x2) = max(x2 + x1) ⊦ x + x3 = [], x3 + x = [], ((max(x + x3) < a0 ⊃ a0;max(x + x3)) < a ⊃ a;(max(x + x3) < a0 ⊃ a0;max(x + x3))) = ((max(x3 + x) < a ⊃ a;max(x3 + x)) < a0 ⊃ a0;(max(x3 + x) < a ⊃ a;max(x3 + x)))

Rule argument

spec: <toplevel> name: max-rec seq: ⊦ x ≠ [] → max(a ' + x) = (max(x) < a ⊃ a;max(x)) subst: [x, a] → [x3 + x, a] flags: <all paths>

Additional information

Used simplifier rules: max-base: ⊦ max(a ') = a;% ⊦ x + y = [] ↔ x = [] ∧ y = []; (as)% ⊦ [] + x = x; (s)% ⊦ x + [] = x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: x3 + x, a