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Proof step 3 from proof ins-01 (previous step)

∀ a, y. sort(ins≤(a, x0) + y) = ins≤(a, sort(x0 + y)) ⊦ sort(ins≤(a, a0 ' + x0) + y) = ins≤(a, ins≤(a0, sort(x0 + y)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ¬ b < a → ins≤(a, b ' + x) = a + b + x
Proof step 5:
¬ ¬ a0 < a, ∀ a, y. sort(ins≤(a, x0) + y) = ins≤(a, sort(x0 + y)) ⊦ sort(ins≤(a, a0 ' + x0) + y) = ins≤(a, ins≤(a0, sort(x0 + y)))

Rule argument

spec: olist inst: name: ins-y seq: ⊦ ¬ b < a → ins≤(a, b ' + x) = a + b + x subst: [x, b, a] → [x0, a0, a] flags: rewrite

Additional information

Used simplifier rules: rec: ⊦ sort(a ' + x) = ins≤(a, sort(x));%c: ⊦ a + x = a ' + x; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%eq: ⊦ ins≤(a, x) = ins≤(a, y) ↔ x = y; from specification olist Used terms: x0, a0, a