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Proof step 16 from proof sort-max-last (previous step)

sort(x1 + a ' + x0) = sort(x1 + x0) + a ', ordered≤(sort(x1 + x0) + a '), a < max(sort(x1 + x0) + a '), max((a ' + x1) + x0) < max(sort(x1 + x0) + a '), max(x1 + x0) < max(sort(x1 + x0) + a ') ⊦ ins≤(a, sort(x1 + x0) + a ') = (sort(x1 + x0) + a ') + a ', a < max((a ' + x1) + x0), a < max(x1 + x0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ x ≠ [] → max(a ' + x) = (max(x) < a ⊃ a;max(x))
Proof step 18:
x1 + x0 ≠ [], sort(x1 + a ' + x0) = sort(x1 + x0) + a ', ordered≤(sort(x1 + x0) + a '), a < max(sort(x1 + x0) + a '), (max(x1 + x0) < a ⊃ a;max(x1 + x0)) < max(sort(x1 + x0) + a '), max(x1 + x0) < max(sort(x1 + x0) + a ') ⊦ ins≤(a, sort(x1 + x0) + a ') = (sort(x1 + x0) + a ') + a ', a < (max(x1 + x0) < a ⊃ a;max(x1 + x0)), a < max(x1 + x0)

Rule argument

spec: inst: name: max-rec seq: ⊦ x ≠ [] → max(a ' + x) = (max(x) < a ⊃ a;max(x)) subst: [x, a] → [x1 + x0, a] flags: rewrite, current

Additional information

Used simplifier rules: app-02-ant: ⊦ ¬ x + y ≠ [] ↔ x = [] ∧ y = []; from specification list%Nil: ⊦ [] + x = x; from specification list%app: ⊦ x + [] = x; from specification list%one: ⊦ sort(a ') = a '; from specification olist-sort%empty: ⊦ sort([]) = []; from specification olist-sort%c-04: ⊦ a ' = b ' ↔ a = b; from specification list%one-02: ⊦ ins≤(a, a ') = a ' + a '; from specification olist%app-08: ⊦ a ' + x = a ' + y ↔ x = y; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: x1 + x0, a