author nipkow Wed, 25 Aug 2021 21:46:34 +0200 changeset 74192 852df4f1dbfa parent 74191 ec947c18e060 child 74193 2b00c267196e
unhide canonical function def examples
 src/HOL/Examples/Functions.thy file | annotate | diff | comparison | revisions src/HOL/ex/Functions.thy file | annotate | diff | comparison | revisions
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Examples/Functions.thy	Wed Aug 25 21:46:34 2021 +0200
@@ -0,0 +1,519 @@
+(*  Title:      HOL/ex/Functions.thy
+    Author:     Alexander Krauss, TU Muenchen
+*)
+
+section \<open>Examples of function definitions\<close>
+
+theory Functions
+begin
+
+subsection \<open>Very basic\<close>
+
+fun fib :: "nat \<Rightarrow> nat"
+where
+  "fib 0 = 1"
+| "fib (Suc 0) = 1"
+| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
+
+text \<open>Partial simp and induction rules:\<close>
+thm fib.psimps
+thm fib.pinduct
+
+text \<open>There is also a cases rule to distinguish cases along the definition:\<close>
+thm fib.cases
+
+
+text \<open>Total simp and induction rules:\<close>
+thm fib.simps
+thm fib.induct
+
+text \<open>Elimination rules:\<close>
+thm fib.elims
+
+
+subsection \<open>Currying\<close>
+
+where
+  "add 0 y = y"
+
+thm add.induct  \<comment> \<open>Note the curried induction predicate\<close>
+
+
+subsection \<open>Nested recursion\<close>
+
+function nz
+where
+  "nz 0 = 0"
+| "nz (Suc x) = nz (nz x)"
+by pat_completeness auto
+
+lemma nz_is_zero:  \<comment> \<open>A lemma we need to prove termination\<close>
+  assumes trm: "nz_dom x"
+  shows "nz x = 0"
+using trm
+by induct (auto simp: nz.psimps)
+
+termination nz
+  by (relation "less_than") (auto simp:nz_is_zero)
+
+thm nz.simps
+thm nz.induct
+
+
+subsubsection \<open>Here comes McCarthy's 91-function\<close>
+
+function f91 :: "nat \<Rightarrow> nat"
+where
+  "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
+by pat_completeness auto
+
+text \<open>Prove a lemma before attempting a termination proof:\<close>
+lemma f91_estimate:
+  assumes trm: "f91_dom n"
+  shows "n < f91 n + 11"
+using trm by induct (auto simp: f91.psimps)
+
+termination
+proof
+  let ?R = "measure (\<lambda>x. 101 - x)"
+  show "wf ?R" ..
+
+  fix n :: nat
+  assume "\<not> 100 < n"  \<comment> \<open>Inner call\<close>
+  then show "(n + 11, n) \<in> ?R" by simp
+
+  assume inner_trm: "f91_dom (n + 11)"  \<comment> \<open>Outer call\<close>
+  with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
+  with \<open>\<not> 100 < n\<close> show "(f91 (n + 11), n) \<in> ?R" by simp
+qed
+
+text \<open>Now trivial (even though it does not belong here):\<close>
+lemma "f91 n = (if 100 < n then n - 10 else 91)"
+  by (induct n rule: f91.induct) auto
+
+
+subsubsection \<open>Here comes Takeuchi's function\<close>
+
+definition tak_m1 where "tak_m1 = (\<lambda>(x,y,z). if x \<le> y then 0 else 1)"
+definition tak_m2 where "tak_m2 = (\<lambda>(x,y,z). nat (Max {x, y, z} - Min {x, y, z}))"
+definition tak_m3 where "tak_m3 = (\<lambda>(x,y,z). nat (x - Min {x, y, z}))"
+
+function tak :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" where
+  "tak x y z = (if x \<le> y then y else tak (tak (x-1) y z) (tak (y-1) z x) (tak (z-1) x y))"
+  by auto
+
+lemma tak_pcorrect:
+  "tak_dom (x, y, z) \<Longrightarrow> tak x y z = (if x \<le> y then y else if y \<le> z then z else x)"
+  by (induction x y z rule: tak.pinduct) (auto simp: tak.psimps)
+
+termination
+  by (relation "tak_m1 <*mlex*> tak_m2 <*mlex*> tak_m3 <*mlex*> {}")
+     (auto simp: mlex_iff wf_mlex tak_pcorrect tak_m1_def tak_m2_def tak_m3_def min_def max_def)
+
+theorem tak_correct: "tak x y z = (if x \<le> y then y else if y \<le> z then z else x)"
+  by (induction x y z rule: tak.induct) auto
+
+
+subsection \<open>More general patterns\<close>
+
+subsubsection \<open>Overlapping patterns\<close>
+
+text \<open>
+  Currently, patterns must always be compatible with each other, since
+  no automatic splitting takes place. But the following definition of
+  GCD is OK, although patterns overlap:
+\<close>
+
+fun gcd2 :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "gcd2 x 0 = x"
+| "gcd2 0 y = y"
+| "gcd2 (Suc x) (Suc y) = (if x < y then gcd2 (Suc x) (y - x)
+                                    else gcd2 (x - y) (Suc y))"
+
+thm gcd2.simps
+thm gcd2.induct
+
+
+subsubsection \<open>Guards\<close>
+
+text \<open>We can reformulate the above example using guarded patterns:\<close>
+
+function gcd3 :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "gcd3 x 0 = x"
+| "gcd3 0 y = y"
+| "gcd3 (Suc x) (Suc y) = gcd3 (Suc x) (y - x)" if "x < y"
+| "gcd3 (Suc x) (Suc y) = gcd3 (x - y) (Suc y)" if "\<not> x < y"
+  apply (case_tac x, case_tac a, auto)
+  apply (case_tac ba, auto)
+  done
+termination by lexicographic_order
+
+thm gcd3.simps
+thm gcd3.induct
+
+
+text \<open>General patterns allow even strange definitions:\<close>
+
+function ev :: "nat \<Rightarrow> bool"
+where
+  "ev (2 * n) = True"
+| "ev (2 * n + 1) = False"
+proof -  \<comment> \<open>completeness is more difficult here \dots\<close>
+  fix P :: bool
+  fix x :: nat
+  assume c1: "\<And>n. x = 2 * n \<Longrightarrow> P"
+    and c2: "\<And>n. x = 2 * n + 1 \<Longrightarrow> P"
+  have divmod: "x = 2 * (x div 2) + (x mod 2)" by auto
+  show P
+  proof (cases "x mod 2 = 0")
+    case True
+    with divmod have "x = 2 * (x div 2)" by simp
+    with c1 show "P" .
+  next
+    case False
+    then have "x mod 2 = 1" by simp
+    with divmod have "x = 2 * (x div 2) + 1" by simp
+    with c2 show "P" .
+  qed
+qed presburger+  \<comment> \<open>solve compatibility with presburger\<close>
+termination by lexicographic_order
+
+thm ev.simps
+thm ev.induct
+thm ev.cases
+
+
+subsection \<open>Mutual Recursion\<close>
+
+fun evn od :: "nat \<Rightarrow> bool"
+where
+  "evn 0 = True"
+| "od 0 = False"
+| "evn (Suc n) = od n"
+| "od (Suc n) = evn n"
+
+thm evn.simps
+thm od.simps
+
+thm evn_od.induct
+thm evn_od.termination
+
+thm evn.elims
+thm od.elims
+
+
+subsection \<open>Definitions in local contexts\<close>
+
+locale my_monoid =
+  fixes opr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+    and un :: "'a"
+  assumes assoc: "opr (opr x y) z = opr x (opr y z)"
+    and lunit: "opr un x = x"
+    and runit: "opr x un = x"
+begin
+
+fun foldR :: "'a list \<Rightarrow> 'a"
+where
+  "foldR [] = un"
+| "foldR (x # xs) = opr x (foldR xs)"
+
+fun foldL :: "'a list \<Rightarrow> 'a"
+where
+  "foldL [] = un"
+| "foldL [x] = x"
+| "foldL (x # y # ys) = foldL (opr x y # ys)"
+
+thm foldL.simps
+
+lemma foldR_foldL: "foldR xs = foldL xs"
+  by (induct xs rule: foldL.induct) (auto simp:lunit runit assoc)
+
+thm foldR_foldL
+
+end
+
+thm my_monoid.foldL.simps
+thm my_monoid.foldR_foldL
+
+
+subsection \<open>\<open>fun_cases\<close>\<close>
+
+subsubsection \<open>Predecessor\<close>
+
+fun pred :: "nat \<Rightarrow> nat"
+where
+  "pred 0 = 0"
+| "pred (Suc n) = n"
+
+thm pred.elims
+
+lemma
+  assumes "pred x = y"
+  obtains "x = 0" "y = 0" | "n" where "x = Suc n" "y = n"
+  by (fact pred.elims[OF assms])
+
+
+text \<open>If the predecessor of a number is 0, that number must be 0 or 1.\<close>
+
+fun_cases pred0E[elim]: "pred n = 0"
+
+lemma "pred n = 0 \<Longrightarrow> n = 0 \<or> n = Suc 0"
+  by (erule pred0E) metis+
+
+text \<open>
+  Other expressions on the right-hand side also work, but whether the
+  generated rule is useful depends on how well the simplifier can
+  simplify it. This example works well:
+\<close>
+
+fun_cases pred42E[elim]: "pred n = 42"
+
+lemma "pred n = 42 \<Longrightarrow> n = 43"
+  by (erule pred42E)
+
+
+subsubsection \<open>List to option\<close>
+
+fun list_to_option :: "'a list \<Rightarrow> 'a option"
+where
+  "list_to_option [x] = Some x"
+| "list_to_option _ = None"
+
+fun_cases list_to_option_NoneE: "list_to_option xs = None"
+  and list_to_option_SomeE: "list_to_option xs = Some x"
+
+lemma "list_to_option xs = Some y \<Longrightarrow> xs = [y]"
+  by (erule list_to_option_SomeE)
+
+
+subsubsection \<open>Boolean Functions\<close>
+
+fun xor :: "bool \<Rightarrow> bool \<Rightarrow> bool"
+where
+  "xor False False = False"
+| "xor True True = False"
+| "xor _ _ = True"
+
+thm xor.elims
+
+text \<open>
+  \<open>fun_cases\<close> does not only recognise function equations, but also works with
+  functions that return a boolean, e.g.:
+\<close>
+
+fun_cases xor_TrueE: "xor a b" and xor_FalseE: "\<not>xor a b"
+print_theorems
+
+
+subsubsection \<open>Many parameters\<close>
+
+fun sum4 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where "sum4 a b c d = a + b + c + d"
+
+fun_cases sum40E: "sum4 a b c d = 0"
+
+lemma "sum4 a b c d = 0 \<Longrightarrow> a = 0"
+  by (erule sum40E)
+
+
+subsection \<open>Partial Function Definitions\<close>
+
+text \<open>Partial functions in the option monad:\<close>
+
+partial_function (option)
+  collatz :: "nat \<Rightarrow> nat list option"
+where
+  "collatz n =
+    (if n \<le> 1 then Some [n]
+     else if even n
+       then do { ns \<leftarrow> collatz (n div 2); Some (n # ns) }
+       else do { ns \<leftarrow> collatz (3 * n + 1);  Some (n # ns)})"
+
+declare collatz.simps[code]
+value "collatz 23"
+
+
+text \<open>Tail-recursive functions:\<close>
+
+partial_function (tailrec) fixpoint :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
+where
+  "fixpoint f x = (if f x = x then x else fixpoint f (f x))"
+
+
+subsection \<open>Regression tests\<close>
+
+text \<open>
+  The following examples mainly serve as tests for the
+  function package.
+\<close>
+
+fun listlen :: "'a list \<Rightarrow> nat"
+where
+  "listlen [] = 0"
+| "listlen (x#xs) = Suc (listlen xs)"
+
+
+subsubsection \<open>Context recursion\<close>
+
+fun f :: "nat \<Rightarrow> nat"
+where
+  zero: "f 0 = 0"
+| succ: "f (Suc n) = (if f n = 0 then 0 else f n)"
+
+
+subsubsection \<open>A combination of context and nested recursion\<close>
+
+function h :: "nat \<Rightarrow> nat"
+where
+  "h 0 = 0"
+| "h (Suc n) = (if h n = 0 then h (h n) else h n)"
+by pat_completeness auto
+
+
+subsubsection \<open>Context, but no recursive call\<close>
+
+fun i :: "nat \<Rightarrow> nat"
+where
+  "i 0 = 0"
+| "i (Suc n) = (if n = 0 then 0 else i n)"
+
+
+subsubsection \<open>Tupled nested recursion\<close>
+
+fun fa :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "fa 0 y = 0"
+| "fa (Suc n) y = (if fa n y = 0 then 0 else fa n y)"
+
+
+subsubsection \<open>Let\<close>
+
+fun j :: "nat \<Rightarrow> nat"
+where
+  "j 0 = 0"
+| "j (Suc n) = (let u = n in Suc (j u))"
+
+
+text \<open>There were some problems with fresh names \dots\<close>
+function  k :: "nat \<Rightarrow> nat"
+where
+  "k x = (let a = x; b = x in k x)"
+  by pat_completeness auto
+
+
+function f2 :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
+where
+  "f2 p = (let (x,y) = p in f2 (y,x))"
+  by pat_completeness auto
+
+
+subsubsection \<open>Abbreviations\<close>
+
+fun f3 :: "'a set \<Rightarrow> bool"
+where
+  "f3 x = finite x"
+
+
+subsubsection \<open>Simple Higher-Order Recursion\<close>
+
+datatype 'a tree = Leaf 'a | Branch "'a tree list"
+
+fun treemap :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a tree \<Rightarrow> 'a tree"
+where
+  "treemap fn (Leaf n) = (Leaf (fn n))"
+| "treemap fn (Branch l) = (Branch (map (treemap fn) l))"
+
+fun tinc :: "nat tree \<Rightarrow> nat tree"
+where
+  "tinc (Leaf n) = Leaf (Suc n)"
+| "tinc (Branch l) = Branch (map tinc l)"
+
+fun testcase :: "'a tree \<Rightarrow> 'a list"
+where
+  "testcase (Leaf a) = [a]"
+| "testcase (Branch x) =
+    (let xs = concat (map testcase x);
+         ys = concat (map testcase x) in
+     xs @ ys)"
+
+
+subsubsection \<open>Pattern matching on records\<close>
+
+record point =
+  Xcoord :: int
+  Ycoord :: int
+
+function swp :: "point \<Rightarrow> point"
+where
+  "swp \<lparr> Xcoord = x, Ycoord = y \<rparr> = \<lparr> Xcoord = y, Ycoord = x \<rparr>"
+proof -
+  fix P x
+  assume "\<And>xa y. x = \<lparr>Xcoord = xa, Ycoord = y\<rparr> \<Longrightarrow> P"
+  then show P by (cases x)
+qed auto
+termination by rule auto
+
+
+subsubsection \<open>The diagonal function\<close>
+
+fun diag :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> nat"
+where
+  "diag x True False = 1"
+| "diag False y True = 2"
+| "diag True False z = 3"
+| "diag True True True = 4"
+| "diag False False False = 5"
+
+
+
+datatype DT =
+  A | B | C | D | E | F | G | H | I | J | K | L | M | N | P
+| Q | R | S | T | U | V
+
+fun big :: "DT \<Rightarrow> nat"
+where
+  "big A = 0"
+| "big B = 0"
+| "big C = 0"
+| "big D = 0"
+| "big E = 0"
+| "big F = 0"
+| "big G = 0"
+| "big H = 0"
+| "big I = 0"
+| "big J = 0"
+| "big K = 0"
+| "big L = 0"
+| "big M = 0"
+| "big N = 0"
+| "big P = 0"
+| "big Q = 0"
+| "big R = 0"
+| "big S = 0"
+| "big T = 0"
+| "big U = 0"
+| "big V = 0"
+
+
+subsubsection \<open>Automatic pattern splitting\<close>
+
+fun f4 :: "nat \<Rightarrow> nat \<Rightarrow> bool"
+where
+  "f4 0 0 = True"
+| "f4 _ _ = False"
+
+
+subsubsection \<open>Polymorphic partial-function\<close>
+
+partial_function (option) f5 :: "'a list \<Rightarrow> 'a option"
+where
+  "f5 x = f5 x"
+
+end```
```--- a/src/HOL/ex/Functions.thy	Wed Aug 25 21:43:43 2021 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,519 +0,0 @@
-(*  Title:      HOL/ex/Functions.thy
-    Author:     Alexander Krauss, TU Muenchen
-*)
-
-section \<open>Examples of function definitions\<close>
-
-theory Functions
-begin
-
-subsection \<open>Very basic\<close>
-
-fun fib :: "nat \<Rightarrow> nat"
-where
-  "fib 0 = 1"
-| "fib (Suc 0) = 1"
-| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
-
-text \<open>Partial simp and induction rules:\<close>
-thm fib.psimps
-thm fib.pinduct
-
-text \<open>There is also a cases rule to distinguish cases along the definition:\<close>
-thm fib.cases
-
-
-text \<open>Total simp and induction rules:\<close>
-thm fib.simps
-thm fib.induct
-
-text \<open>Elimination rules:\<close>
-thm fib.elims
-
-
-subsection \<open>Currying\<close>
-
-where
-  "add 0 y = y"
-
-thm add.induct  \<comment> \<open>Note the curried induction predicate\<close>
-
-
-subsection \<open>Nested recursion\<close>
-
-function nz
-where
-  "nz 0 = 0"
-| "nz (Suc x) = nz (nz x)"
-by pat_completeness auto
-
-lemma nz_is_zero:  \<comment> \<open>A lemma we need to prove termination\<close>
-  assumes trm: "nz_dom x"
-  shows "nz x = 0"
-using trm
-by induct (auto simp: nz.psimps)
-
-termination nz
-  by (relation "less_than") (auto simp:nz_is_zero)
-
-thm nz.simps
-thm nz.induct
-
-
-subsubsection \<open>Here comes McCarthy's 91-function\<close>
-
-function f91 :: "nat \<Rightarrow> nat"
-where
-  "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
-by pat_completeness auto
-
-text \<open>Prove a lemma before attempting a termination proof:\<close>
-lemma f91_estimate:
-  assumes trm: "f91_dom n"
-  shows "n < f91 n + 11"
-using trm by induct (auto simp: f91.psimps)
-
-termination
-proof
-  let ?R = "measure (\<lambda>x. 101 - x)"
-  show "wf ?R" ..
-
-  fix n :: nat
-  assume "\<not> 100 < n"  \<comment> \<open>Inner call\<close>
-  then show "(n + 11, n) \<in> ?R" by simp
-
-  assume inner_trm: "f91_dom (n + 11)"  \<comment> \<open>Outer call\<close>
-  with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
-  with \<open>\<not> 100 < n\<close> show "(f91 (n + 11), n) \<in> ?R" by simp
-qed
-
-text \<open>Now trivial (even though it does not belong here):\<close>
-lemma "f91 n = (if 100 < n then n - 10 else 91)"
-  by (induct n rule: f91.induct) auto
-
-
-subsubsection \<open>Here comes Takeuchi's function\<close>
-
-definition tak_m1 where "tak_m1 = (\<lambda>(x,y,z). if x \<le> y then 0 else 1)"
-definition tak_m2 where "tak_m2 = (\<lambda>(x,y,z). nat (Max {x, y, z} - Min {x, y, z}))"
-definition tak_m3 where "tak_m3 = (\<lambda>(x,y,z). nat (x - Min {x, y, z}))"
-
-function tak :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" where
-  "tak x y z = (if x \<le> y then y else tak (tak (x-1) y z) (tak (y-1) z x) (tak (z-1) x y))"
-  by auto
-
-lemma tak_pcorrect:
-  "tak_dom (x, y, z) \<Longrightarrow> tak x y z = (if x \<le> y then y else if y \<le> z then z else x)"
-  by (induction x y z rule: tak.pinduct) (auto simp: tak.psimps)
-
-termination
-  by (relation "tak_m1 <*mlex*> tak_m2 <*mlex*> tak_m3 <*mlex*> {}")
-     (auto simp: mlex_iff wf_mlex tak_pcorrect tak_m1_def tak_m2_def tak_m3_def min_def max_def)
-
-theorem tak_correct: "tak x y z = (if x \<le> y then y else if y \<le> z then z else x)"
-  by (induction x y z rule: tak.induct) auto
-
-
-subsection \<open>More general patterns\<close>
-
-subsubsection \<open>Overlapping patterns\<close>
-
-text \<open>
-  Currently, patterns must always be compatible with each other, since
-  no automatic splitting takes place. But the following definition of
-  GCD is OK, although patterns overlap:
-\<close>
-
-fun gcd2 :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "gcd2 x 0 = x"
-| "gcd2 0 y = y"
-| "gcd2 (Suc x) (Suc y) = (if x < y then gcd2 (Suc x) (y - x)
-                                    else gcd2 (x - y) (Suc y))"
-
-thm gcd2.simps
-thm gcd2.induct
-
-
-subsubsection \<open>Guards\<close>
-
-text \<open>We can reformulate the above example using guarded patterns:\<close>
-
-function gcd3 :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "gcd3 x 0 = x"
-| "gcd3 0 y = y"
-| "gcd3 (Suc x) (Suc y) = gcd3 (Suc x) (y - x)" if "x < y"
-| "gcd3 (Suc x) (Suc y) = gcd3 (x - y) (Suc y)" if "\<not> x < y"
-  apply (case_tac x, case_tac a, auto)
-  apply (case_tac ba, auto)
-  done
-termination by lexicographic_order
-
-thm gcd3.simps
-thm gcd3.induct
-
-
-text \<open>General patterns allow even strange definitions:\<close>
-
-function ev :: "nat \<Rightarrow> bool"
-where
-  "ev (2 * n) = True"
-| "ev (2 * n + 1) = False"
-proof -  \<comment> \<open>completeness is more difficult here \dots\<close>
-  fix P :: bool
-  fix x :: nat
-  assume c1: "\<And>n. x = 2 * n \<Longrightarrow> P"
-    and c2: "\<And>n. x = 2 * n + 1 \<Longrightarrow> P"
-  have divmod: "x = 2 * (x div 2) + (x mod 2)" by auto
-  show P
-  proof (cases "x mod 2 = 0")
-    case True
-    with divmod have "x = 2 * (x div 2)" by simp
-    with c1 show "P" .
-  next
-    case False
-    then have "x mod 2 = 1" by simp
-    with divmod have "x = 2 * (x div 2) + 1" by simp
-    with c2 show "P" .
-  qed
-qed presburger+  \<comment> \<open>solve compatibility with presburger\<close>
-termination by lexicographic_order
-
-thm ev.simps
-thm ev.induct
-thm ev.cases
-
-
-subsection \<open>Mutual Recursion\<close>
-
-fun evn od :: "nat \<Rightarrow> bool"
-where
-  "evn 0 = True"
-| "od 0 = False"
-| "evn (Suc n) = od n"
-| "od (Suc n) = evn n"
-
-thm evn.simps
-thm od.simps
-
-thm evn_od.induct
-thm evn_od.termination
-
-thm evn.elims
-thm od.elims
-
-
-subsection \<open>Definitions in local contexts\<close>
-
-locale my_monoid =
-  fixes opr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
-    and un :: "'a"
-  assumes assoc: "opr (opr x y) z = opr x (opr y z)"
-    and lunit: "opr un x = x"
-    and runit: "opr x un = x"
-begin
-
-fun foldR :: "'a list \<Rightarrow> 'a"
-where
-  "foldR [] = un"
-| "foldR (x # xs) = opr x (foldR xs)"
-
-fun foldL :: "'a list \<Rightarrow> 'a"
-where
-  "foldL [] = un"
-| "foldL [x] = x"
-| "foldL (x # y # ys) = foldL (opr x y # ys)"
-
-thm foldL.simps
-
-lemma foldR_foldL: "foldR xs = foldL xs"
-  by (induct xs rule: foldL.induct) (auto simp:lunit runit assoc)
-
-thm foldR_foldL
-
-end
-
-thm my_monoid.foldL.simps
-thm my_monoid.foldR_foldL
-
-
-subsection \<open>\<open>fun_cases\<close>\<close>
-
-subsubsection \<open>Predecessor\<close>
-
-fun pred :: "nat \<Rightarrow> nat"
-where
-  "pred 0 = 0"
-| "pred (Suc n) = n"
-
-thm pred.elims
-
-lemma
-  assumes "pred x = y"
-  obtains "x = 0" "y = 0" | "n" where "x = Suc n" "y = n"
-  by (fact pred.elims[OF assms])
-
-
-text \<open>If the predecessor of a number is 0, that number must be 0 or 1.\<close>
-
-fun_cases pred0E[elim]: "pred n = 0"
-
-lemma "pred n = 0 \<Longrightarrow> n = 0 \<or> n = Suc 0"
-  by (erule pred0E) metis+
-
-text \<open>
-  Other expressions on the right-hand side also work, but whether the
-  generated rule is useful depends on how well the simplifier can
-  simplify it. This example works well:
-\<close>
-
-fun_cases pred42E[elim]: "pred n = 42"
-
-lemma "pred n = 42 \<Longrightarrow> n = 43"
-  by (erule pred42E)
-
-
-subsubsection \<open>List to option\<close>
-
-fun list_to_option :: "'a list \<Rightarrow> 'a option"
-where
-  "list_to_option [x] = Some x"
-| "list_to_option _ = None"
-
-fun_cases list_to_option_NoneE: "list_to_option xs = None"
-  and list_to_option_SomeE: "list_to_option xs = Some x"
-
-lemma "list_to_option xs = Some y \<Longrightarrow> xs = [y]"
-  by (erule list_to_option_SomeE)
-
-
-subsubsection \<open>Boolean Functions\<close>
-
-fun xor :: "bool \<Rightarrow> bool \<Rightarrow> bool"
-where
-  "xor False False = False"
-| "xor True True = False"
-| "xor _ _ = True"
-
-thm xor.elims
-
-text \<open>
-  \<open>fun_cases\<close> does not only recognise function equations, but also works with
-  functions that return a boolean, e.g.:
-\<close>
-
-fun_cases xor_TrueE: "xor a b" and xor_FalseE: "\<not>xor a b"
-print_theorems
-
-
-subsubsection \<open>Many parameters\<close>
-
-fun sum4 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-  where "sum4 a b c d = a + b + c + d"
-
-fun_cases sum40E: "sum4 a b c d = 0"
-
-lemma "sum4 a b c d = 0 \<Longrightarrow> a = 0"
-  by (erule sum40E)
-
-
-subsection \<open>Partial Function Definitions\<close>
-
-text \<open>Partial functions in the option monad:\<close>
-
-partial_function (option)
-  collatz :: "nat \<Rightarrow> nat list option"
-where
-  "collatz n =
-    (if n \<le> 1 then Some [n]
-     else if even n
-       then do { ns \<leftarrow> collatz (n div 2); Some (n # ns) }
-       else do { ns \<leftarrow> collatz (3 * n + 1);  Some (n # ns)})"
-
-declare collatz.simps[code]
-value "collatz 23"
-
-
-text \<open>Tail-recursive functions:\<close>
-
-partial_function (tailrec) fixpoint :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
-where
-  "fixpoint f x = (if f x = x then x else fixpoint f (f x))"
-
-
-subsection \<open>Regression tests\<close>
-
-text \<open>
-  The following examples mainly serve as tests for the
-  function package.
-\<close>
-
-fun listlen :: "'a list \<Rightarrow> nat"
-where
-  "listlen [] = 0"
-| "listlen (x#xs) = Suc (listlen xs)"
-
-
-subsubsection \<open>Context recursion\<close>
-
-fun f :: "nat \<Rightarrow> nat"
-where
-  zero: "f 0 = 0"
-| succ: "f (Suc n) = (if f n = 0 then 0 else f n)"
-
-
-subsubsection \<open>A combination of context and nested recursion\<close>
-
-function h :: "nat \<Rightarrow> nat"
-where
-  "h 0 = 0"
-| "h (Suc n) = (if h n = 0 then h (h n) else h n)"
-by pat_completeness auto
-
-
-subsubsection \<open>Context, but no recursive call\<close>
-
-fun i :: "nat \<Rightarrow> nat"
-where
-  "i 0 = 0"
-| "i (Suc n) = (if n = 0 then 0 else i n)"
-
-
-subsubsection \<open>Tupled nested recursion\<close>
-
-fun fa :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "fa 0 y = 0"
-| "fa (Suc n) y = (if fa n y = 0 then 0 else fa n y)"
-
-
-subsubsection \<open>Let\<close>
-
-fun j :: "nat \<Rightarrow> nat"
-where
-  "j 0 = 0"
-| "j (Suc n) = (let u = n in Suc (j u))"
-
-
-text \<open>There were some problems with fresh names \dots\<close>
-function  k :: "nat \<Rightarrow> nat"
-where
-  "k x = (let a = x; b = x in k x)"
-  by pat_completeness auto
-
-
-function f2 :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
-where
-  "f2 p = (let (x,y) = p in f2 (y,x))"
-  by pat_completeness auto
-
-
-subsubsection \<open>Abbreviations\<close>
-
-fun f3 :: "'a set \<Rightarrow> bool"
-where
-  "f3 x = finite x"
-
-
-subsubsection \<open>Simple Higher-Order Recursion\<close>
-
-datatype 'a tree = Leaf 'a | Branch "'a tree list"
-
-fun treemap :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a tree \<Rightarrow> 'a tree"
-where
-  "treemap fn (Leaf n) = (Leaf (fn n))"
-| "treemap fn (Branch l) = (Branch (map (treemap fn) l))"
-
-fun tinc :: "nat tree \<Rightarrow> nat tree"
-where
-  "tinc (Leaf n) = Leaf (Suc n)"
-| "tinc (Branch l) = Branch (map tinc l)"
-
-fun testcase :: "'a tree \<Rightarrow> 'a list"
-where
-  "testcase (Leaf a) = [a]"
-| "testcase (Branch x) =
-    (let xs = concat (map testcase x);
-         ys = concat (map testcase x) in
-     xs @ ys)"
-
-
-subsubsection \<open>Pattern matching on records\<close>
-
-record point =
-  Xcoord :: int
-  Ycoord :: int
-
-function swp :: "point \<Rightarrow> point"
-where
-  "swp \<lparr> Xcoord = x, Ycoord = y \<rparr> = \<lparr> Xcoord = y, Ycoord = x \<rparr>"
-proof -
-  fix P x
-  assume "\<And>xa y. x = \<lparr>Xcoord = xa, Ycoord = y\<rparr> \<Longrightarrow> P"
-  then show P by (cases x)
-qed auto
-termination by rule auto
-
-
-subsubsection \<open>The diagonal function\<close>
-
-fun diag :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> nat"
-where
-  "diag x True False = 1"
-| "diag False y True = 2"
-| "diag True False z = 3"
-| "diag True True True = 4"
-| "diag False False False = 5"
-
-
-
-datatype DT =
-  A | B | C | D | E | F | G | H | I | J | K | L | M | N | P
-| Q | R | S | T | U | V
-
-fun big :: "DT \<Rightarrow> nat"
-where
-  "big A = 0"
-| "big B = 0"
-| "big C = 0"
-| "big D = 0"
-| "big E = 0"
-| "big F = 0"
-| "big G = 0"
-| "big H = 0"
-| "big I = 0"
-| "big J = 0"
-| "big K = 0"
-| "big L = 0"
-| "big M = 0"
-| "big N = 0"
-| "big P = 0"
-| "big Q = 0"
-| "big R = 0"
-| "big S = 0"
-| "big T = 0"
-| "big U = 0"
-| "big V = 0"
-
-
-subsubsection \<open>Automatic pattern splitting\<close>
-
-fun f4 :: "nat \<Rightarrow> nat \<Rightarrow> bool"
-where
-  "f4 0 0 = True"
-| "f4 _ _ = False"
-
-
-subsubsection \<open>Polymorphic partial-function\<close>
-
-partial_function (option) f5 :: "'a list \<Rightarrow> 'a option"
-where
-  "f5 x = f5 x"
-
-end```