| products/Sources/formale Sprachen/Isabelle/HOL/Data_Structures/ (Netbeans IDE Version 28©) Datei vom 16.11.2025 mit Größe 2 kB |
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Impressum Sorted_Less.thy Sprache: Isabelle |
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section <<close theory Sorted_Less imports Less_False begin hide_const sorted text ‹Is a list sorted without duplicates, i.e., wrt ‹<\< sorted :: "'a::linorder list → bool" where sorted ≡ sorted_wrt (<)" sorted_wrt_Cons = sorted_wrt.simps(2) ‹The definition of const‹sorted_wrt› relates each element to all the elements a causes a blowup of the formulas. Thus we simplify matters by only comparing adj sorted_wrt.simps(2)[simp del] sorted_wrt1[simp] sorted_wrt2[OF transp_on_less, simp] sorted_cons: "sorted (x#xs) ==> sorted xs" (simp add: sorted_wrt_Cons) sorted_cons': "ASSUMPTION (sorted (x#xs)) ==> sorted xs" (rule ASSUMPTION_D [THEN sorted_cons]) sorted_snoc: "sorted (xs @ [y]) ==> sorted xs" (simp add: sorted_wrt_append) sorted_snoc': "ASSUMPTION (sorted (xs @ [y])) ==> sorted xs" (rule ASSUMPTION_D [THEN sorted_snoc]) sorted_mid_iff: "sorted(xs @ y # ys) = (sorted(xs @ [y]) ∧ sorted(y # ys))" (fastforce simp add: sorted_wrt_Cons sorted_wrt_append) sorted_mid_iff2: "sorted(x # xs @ y # ys) = (sorted(x # xs) ∧ x < y ∧ sorted(xs @ [y]) ∧ sorted(y # ys))" (fastforce simp add: sorted_wrt_Cons sorted_wrt_append) sorted_mid_iff': "NO_MATCH [] ys ==> sorted(xs @ y # ys) = (sorted(xs @ [y]) ∧ sorted(y # ys))" (rule sorted_mid_iff) sorted_lems = sorted_mid_iff' sorted_mid_iff2 sorted_cons' sorted_snoc' ‹Splay trees need two additional const‹sorted› lemmas:› sorted_snoc_le: "ASSUMPTION(sorted(xs @ [x])) ==> x ≤ y ==> sorted (xs @ [y])" (auto simp add: sorted_wrt_append ASSUMPTION_def) sorted_Cons_le: "ASSUMPTION(sorted(x # xs)) ==> y ≤ x ==> sorted (y # xs)" (auto simp add: sorted_wrt_Cons ASSUMPTION_def)
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2026-10-11