# Theory Sorted_Less

theory Sorted_Less
imports Less_False
```(* Author: Tobias Nipkow *)

section ‹Lists Sorted wrt \$<\$›

theory Sorted_Less
imports Less_False
begin

hide_const sorted

text ‹Is a list sorted without duplicates, i.e., wrt @{text"<"}?.›

abbreviation sorted :: "'a::linorder list ⇒ bool" where
"sorted ≡ sorted_wrt (<)"

lemmas sorted_wrt_Cons = sorted_wrt.simps(2)

text ‹The definition of @{const sorted_wrt} relates each element to all the elements after it.
This causes a blowup of the formulas. Thus we simplify matters by only comparing adjacent elements.›

declare
sorted_wrt.simps(2)[simp del]
sorted_wrt1[simp] sorted_wrt2[OF transp_less, simp]

lemma sorted_cons: "sorted (x#xs) ⟹ sorted xs"

lemma sorted_cons': "ASSUMPTION (sorted (x#xs)) ⟹ sorted xs"
by(rule ASSUMPTION_D [THEN sorted_cons])

lemma sorted_snoc: "sorted (xs @ [y]) ⟹ sorted xs"

lemma sorted_snoc': "ASSUMPTION (sorted (xs @ [y])) ⟹ sorted xs"
by(rule ASSUMPTION_D [THEN sorted_snoc])

lemma sorted_mid_iff:
"sorted(xs @ y # ys) = (sorted(xs @ [y]) ∧ sorted(y # ys))"

lemma sorted_mid_iff2:
"sorted(x # xs @ y # ys) =
(sorted(x # xs) ∧ x < y ∧ sorted(xs @ [y]) ∧ sorted(y # ys))"

lemma sorted_mid_iff': "NO_MATCH [] ys ⟹
sorted(xs @ y # ys) = (sorted(xs @ [y]) ∧ sorted(y # ys))"
by(rule sorted_mid_iff)

lemmas sorted_lems = sorted_mid_iff' sorted_mid_iff2 sorted_cons' sorted_snoc'

text‹Splay trees need two additional @{const sorted} lemmas:›

lemma sorted_snoc_le:
"ASSUMPTION(sorted(xs @ [x])) ⟹ x ≤ y ⟹ sorted (xs @ [y])"
by (auto simp add: sorted_wrt_append ASSUMPTION_def)

lemma sorted_Cons_le:
"ASSUMPTION(sorted(x # xs)) ⟹ y ≤ x ⟹ sorted (y # xs)"
by (auto simp add: sorted_wrt_Cons ASSUMPTION_def)

end
```