public_html: Nominal/activities/cas09/Example.thy@5eaec0f9980f (annotated)

415 f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1	theory Example
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2	imports Main
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	3	begin
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	4
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	5	text {*
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	6	A small example theory proving a few facts
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	7	about list append and list reverse.
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	8	*}
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	9
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	10	datatype 'a list2 =
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	11	nil2 ("[]")
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	12	\| cons2 "'a" "'a list2" ("_ ## _")
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	13
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	14	abbreviation
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	15	singleton :: "'a \<Rightarrow> 'a list2" ("[_]")
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	16	where
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	17	"[x] \<equiv> x##[]"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	18
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	19	fun
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	20	append2 :: "'a list2 \<Rightarrow> 'a list2 \<Rightarrow> 'a list2" ("_ @@ _")
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	21	where
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	22	append2_nil2: "[] @@ ys = ys"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	23	\| append2_cons2: "(x##xs) @@ ys = x##(xs @@ ys)"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	24
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	25	fun
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	26	rev2 :: "'a list2 \<Rightarrow> 'a list2"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	27	where
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	28	"rev2 [] = []"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	29	\| "rev2 (x##xs) = (rev2 xs) @@ (x##[])"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	30
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	31	lemma append2_nil2R[simp]:
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	32	shows "xs @@ [] = xs"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	33	by (induct xs) (auto)
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	34
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	35	lemma append2_assoc[simp]:
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	36	shows "(xs @@ ys) @@ zs = xs @@ (ys @@ zs)"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	37	by (induct xs) (auto)
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	38
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	39	lemma rev2_append2[simp]:
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	40	shows "rev2 (xs @@ ys) = (rev2 ys) @@ (rev2 xs)"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	41	by (induct xs) (auto)
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	42
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	43	lemma rev2_rev2:
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	44	shows "rev2 (rev2 xs) = xs"
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	45	by (induct xs) (auto)
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	46
f1be8028a4a9 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	47	end

author	Christian Urban <urbanc@in.tum.de>
	Thu, 13 Sep 2018 13:09:24 +0100
changeset 539	5eaec0f9980f
parent 415	f1be8028a4a9
permissions	-rw-r--r--