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Proof step 19 from proof eq (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ x ≠ [] ∧ x.first < a → ins≤(a, x) = x.first + ins≤(a, x.rest)
Proof step 21:
¬(y ≠ [] ∧ y.first < a), a ' + a0 ' + x0 = ins≤(a, y), ∀ a, y. ins≤(a, x0) = ins≤(a, y) → x0 = y ⊦ a0 ' + x0 = y, a0 < a

Rule argument

spec: olist inst: name: ins-rec seq: ⊦ x ≠ [] ∧ x.first < a → ins≤(a, x) = x.first + ins≤(a, x.rest) subst: [x, a] → [y, a] flags: rewrite, current

Additional information

Used simplifier rules: c: ⊦ a + x = a ' + x; from specification list%c-06: ⊦ a ' + x = b ' + y ↔ a = b ∧ x = y; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list%lel: ⊦ a < b ∧ a = c → c < b; (forward) from specification oelem%nen: ⊦ ¬ b < a ∧ a = c → ¬ b < c; (forward) from specification oelem%irreflexivity: ⊦ ¬ a < a; from specification oelem Used terms: y, a