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Proof step 7 from proof ins-max (previous step)

ordered≤(a0 ' + x), ∀ a1. ins≤(a1, x) = x + a1 ' ∨ a1 < max(x) ⊦ ins≤(a, a0 ' + x) = (a0 ' + x) + a ', a < max(a0 ' + x), ordered≤(a ' + a0 ' + x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ b < a → ins≤(a, b ' + x) = b + ins≤(a, x)
Proof step 9:
a0 < a, ordered≤(a0 ' + x), ∀ a1. ins≤(a1, x) = x + a1 ' ∨ a1 < max(x) ⊦ a0 + ins≤(a, x) = (a0 ' + x) + a ', a < max(a0 ' + x), ordered≤(a ' + a0 ' + x)

Rule argument

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

Additional information

Used simplifier rules: ins-y: ⊦ ¬ b < a → ins≤(a, b ' + x) = a + b + x; from specification olist%c: ⊦ a + x = a ' + x; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%le-r: ⊦ ordered≤(a ' + b ' + x) ↔ ¬ b < a ∧ ordered≤(b + x); from specification olist Used terms: x, a0, a