author | wenzelm |
Thu, 30 May 2013 20:38:50 +0200 | |
changeset 52252 | 81fcc11d8c65 |
parent 45008 | 8b74cfea913a |
child 53611 | 437c0a63bb16 |
permissions | -rw-r--r-- |
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(* Title: HOL/ex/Fundefs.thy |
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Author: Alexander Krauss, TU Muenchen |
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*) |
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header {* Examples of function definitions *} |
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theory Fundefs |
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imports Parity "~~/src/HOL/Library/Monad_Syntax" |
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begin |
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subsection {* Very basic *} |
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fun fib :: "nat \<Rightarrow> nat" |
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where |
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"fib 0 = 1" |
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| "fib (Suc 0) = 1" |
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| "fib (Suc (Suc n)) = fib n + fib (Suc n)" |
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text {* partial simp and induction rules: *} |
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thm fib.psimps |
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thm fib.pinduct |
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text {* There is also a cases rule to distinguish cases along the definition *} |
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thm fib.cases |
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text {* total simp and induction rules: *} |
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thm fib.simps |
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thm fib.induct |
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subsection {* Currying *} |
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fun add |
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where |
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"add 0 y = y" |
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| "add (Suc x) y = Suc (add x y)" |
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thm add.simps |
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thm add.induct -- {* Note the curried induction predicate *} |
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subsection {* Nested recursion *} |
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function nz |
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where |
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"nz 0 = 0" |
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| "nz (Suc x) = nz (nz x)" |
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by pat_completeness auto |
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lemma nz_is_zero: -- {* A lemma we need to prove termination *} |
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assumes trm: "nz_dom x" |
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shows "nz x = 0" |
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using trm |
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by induct (auto simp: nz.psimps) |
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termination nz |
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by (relation "less_than") (auto simp:nz_is_zero) |
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thm nz.simps |
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thm nz.induct |
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text {* Here comes McCarthy's 91-function *} |
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function f91 :: "nat => nat" |
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where |
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"f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))" |
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by pat_completeness auto |
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(* Prove a lemma before attempting a termination proof *) |
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lemma f91_estimate: |
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assumes trm: "f91_dom n" |
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shows "n < f91 n + 11" |
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using trm by induct (auto simp: f91.psimps) |
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termination |
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proof |
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let ?R = "measure (%x. 101 - x)" |
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show "wf ?R" .. |
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fix n::nat assume "~ 100 < n" (* Inner call *) |
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thus "(n + 11, n) : ?R" by simp |
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assume inner_trm: "f91_dom (n + 11)" (* Outer call *) |
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with f91_estimate have "n + 11 < f91 (n + 11) + 11" . |
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with `~ 100 < n` show "(f91 (n + 11), n) : ?R" by simp |
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qed |
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text{* Now trivial (even though it does not belong here): *} |
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lemma "f91 n = (if 100 < n then n - 10 else 91)" |
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by (induct n rule:f91.induct) auto |
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subsection {* More general patterns *} |
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subsubsection {* Overlapping patterns *} |
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text {* Currently, patterns must always be compatible with each other, since |
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no automatic splitting takes place. But the following definition of |
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gcd is ok, although patterns overlap: *} |
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fun gcd2 :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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where |
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"gcd2 x 0 = x" |
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| "gcd2 0 y = y" |
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| "gcd2 (Suc x) (Suc y) = (if x < y then gcd2 (Suc x) (y - x) |
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else gcd2 (x - y) (Suc y))" |
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thm gcd2.simps |
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thm gcd2.induct |
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subsubsection {* Guards *} |
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text {* We can reformulate the above example using guarded patterns *} |
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function gcd3 :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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where |
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"gcd3 x 0 = x" |
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| "gcd3 0 y = y" |
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| "x < y \<Longrightarrow> gcd3 (Suc x) (Suc y) = gcd3 (Suc x) (y - x)" |
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| "\<not> x < y \<Longrightarrow> gcd3 (Suc x) (Suc y) = gcd3 (x - y) (Suc y)" |
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apply (case_tac x, case_tac a, auto) |
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apply (case_tac ba, auto) |
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done |
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termination by lexicographic_order |
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thm gcd3.simps |
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thm gcd3.induct |
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text {* General patterns allow even strange definitions: *} |
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function ev :: "nat \<Rightarrow> bool" |
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where |
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"ev (2 * n) = True" |
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| "ev (2 * n + 1) = False" |
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proof - -- {* completeness is more difficult here \dots *} |
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fix P :: bool |
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and x :: nat |
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assume c1: "\<And>n. x = 2 * n \<Longrightarrow> P" |
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and c2: "\<And>n. x = 2 * n + 1 \<Longrightarrow> P" |
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have divmod: "x = 2 * (x div 2) + (x mod 2)" by auto |
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show "P" |
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proof cases |
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assume "x mod 2 = 0" |
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with divmod have "x = 2 * (x div 2)" by simp |
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with c1 show "P" . |
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next |
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assume "x mod 2 \<noteq> 0" |
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hence "x mod 2 = 1" by simp |
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with divmod have "x = 2 * (x div 2) + 1" by simp |
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with c2 show "P" . |
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qed |
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qed presburger+ -- {* solve compatibility with presburger *} |
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termination by lexicographic_order |
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thm ev.simps |
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thm ev.induct |
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thm ev.cases |
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subsection {* Mutual Recursion *} |
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fun evn od :: "nat \<Rightarrow> bool" |
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where |
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"evn 0 = True" |
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| "od 0 = False" |
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| "evn (Suc n) = od n" |
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| "od (Suc n) = evn n" |
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thm evn.simps |
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thm od.simps |
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thm evn_od.induct |
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thm evn_od.termination |
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subsection {* Definitions in local contexts *} |
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locale my_monoid = |
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fixes opr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
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and un :: "'a" |
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assumes assoc: "opr (opr x y) z = opr x (opr y z)" |
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and lunit: "opr un x = x" |
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and runit: "opr x un = x" |
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begin |
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fun foldR :: "'a list \<Rightarrow> 'a" |
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where |
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"foldR [] = un" |
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| "foldR (x#xs) = opr x (foldR xs)" |
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fun foldL :: "'a list \<Rightarrow> 'a" |
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where |
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"foldL [] = un" |
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| "foldL [x] = x" |
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| "foldL (x#y#ys) = foldL (opr x y # ys)" |
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thm foldL.simps |
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lemma foldR_foldL: "foldR xs = foldL xs" |
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by (induct xs rule: foldL.induct) (auto simp:lunit runit assoc) |
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thm foldR_foldL |
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end |
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thm my_monoid.foldL.simps |
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thm my_monoid.foldR_foldL |
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subsection {* Partial Function Definitions *} |
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text {* Partial functions in the option monad: *} |
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partial_function (option) |
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collatz :: "nat \<Rightarrow> nat list option" |
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where |
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"collatz n = |
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(if n \<le> 1 then Some [n] |
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else if even n |
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then do { ns \<leftarrow> collatz (n div 2); Some (n # ns) } |
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else do { ns \<leftarrow> collatz (3 * n + 1); Some (n # ns)})" |
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declare collatz.simps[code] |
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value "collatz 23" |
227 |
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text {* Tail-recursive functions: *} |
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partial_function (tailrec) fixpoint :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" |
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where |
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"fixpoint f x = (if f x = x then x else fixpoint f (f x))" |
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subsection {* Regression tests *} |
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text {* The following examples mainly serve as tests for the |
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function package *} |
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fun listlen :: "'a list \<Rightarrow> nat" |
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where |
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"listlen [] = 0" |
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| "listlen (x#xs) = Suc (listlen xs)" |
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(* Context recursion *) |
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fun f :: "nat \<Rightarrow> nat" |
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where |
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zero: "f 0 = 0" |
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| succ: "f (Suc n) = (if f n = 0 then 0 else f n)" |
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(* A combination of context and nested recursion *) |
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function h :: "nat \<Rightarrow> nat" |
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where |
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"h 0 = 0" |
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| "h (Suc n) = (if h n = 0 then h (h n) else h n)" |
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by pat_completeness auto |
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(* Context, but no recursive call: *) |
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fun i :: "nat \<Rightarrow> nat" |
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where |
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"i 0 = 0" |
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| "i (Suc n) = (if n = 0 then 0 else i n)" |
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||
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(* Tupled nested recursion *) |
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fun fa :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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where |
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"fa 0 y = 0" |
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| "fa (Suc n) y = (if fa n y = 0 then 0 else fa n y)" |
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(* Let *) |
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fun j :: "nat \<Rightarrow> nat" |
|
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where |
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"j 0 = 0" |
|
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| "j (Suc n) = (let u = n in Suc (j u))" |
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||
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(* There were some problems with fresh names\<dots> *) |
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function k :: "nat \<Rightarrow> nat" |
|
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where |
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"k x = (let a = x; b = x in k x)" |
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by pat_completeness auto |
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||
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||
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function f2 :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)" |
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where |
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"f2 p = (let (x,y) = p in f2 (y,x))" |
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by pat_completeness auto |
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||
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||
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(* abbreviations *) |
|
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fun f3 :: "'a set \<Rightarrow> bool" |
|
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where |
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"f3 x = finite x" |
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(* Simple Higher-Order Recursion *) |
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datatype 'a tree = |
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Leaf 'a |
|
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| Branch "'a tree list" |
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|
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fun treemap :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a tree \<Rightarrow> 'a tree" |
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where |
308 |
"treemap fn (Leaf n) = (Leaf (fn n))" |
|
309 |
| "treemap fn (Branch l) = (Branch (map (treemap fn) l))" |
|
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311 |
fun tinc :: "nat tree \<Rightarrow> nat tree" |
|
312 |
where |
|
313 |
"tinc (Leaf n) = Leaf (Suc n)" |
|
314 |
| "tinc (Branch l) = Branch (map tinc l)" |
|
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||
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fun testcase :: "'a tree \<Rightarrow> 'a list" |
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|
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where |
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"testcase (Leaf a) = [a]" |
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| "testcase (Branch x) = |
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|
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(let xs = concat (map testcase x); |
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ys = concat (map testcase x) in |
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|
322 |
xs @ ys)" |
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|
323 |
|
22726 | 324 |
|
325 |
(* Pattern matching on records *) |
|
326 |
record point = |
|
327 |
Xcoord :: int |
|
328 |
Ycoord :: int |
|
329 |
||
330 |
function swp :: "point \<Rightarrow> point" |
|
331 |
where |
|
332 |
"swp \<lparr> Xcoord = x, Ycoord = y \<rparr> = \<lparr> Xcoord = y, Ycoord = x \<rparr>" |
|
333 |
proof - |
|
334 |
fix P x |
|
335 |
assume "\<And>xa y. x = \<lparr>Xcoord = xa, Ycoord = y\<rparr> \<Longrightarrow> P" |
|
336 |
thus "P" |
|
337 |
by (cases x) |
|
338 |
qed auto |
|
339 |
termination by rule auto |
|
340 |
||
341 |
||
342 |
(* The diagonal function *) |
|
343 |
fun diag :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> nat" |
|
344 |
where |
|
345 |
"diag x True False = 1" |
|
346 |
| "diag False y True = 2" |
|
347 |
| "diag True False z = 3" |
|
348 |
| "diag True True True = 4" |
|
349 |
| "diag False False False = 5" |
|
350 |
||
351 |
||
352 |
(* Many equations (quadratic blowup) *) |
|
353 |
datatype DT = |
|
354 |
A | B | C | D | E | F | G | H | I | J | K | L | M | N | P |
|
355 |
| Q | R | S | T | U | V |
|
356 |
||
357 |
fun big :: "DT \<Rightarrow> nat" |
|
358 |
where |
|
359 |
"big A = 0" |
|
360 |
| "big B = 0" |
|
361 |
| "big C = 0" |
|
362 |
| "big D = 0" |
|
363 |
| "big E = 0" |
|
364 |
| "big F = 0" |
|
365 |
| "big G = 0" |
|
366 |
| "big H = 0" |
|
367 |
| "big I = 0" |
|
368 |
| "big J = 0" |
|
369 |
| "big K = 0" |
|
370 |
| "big L = 0" |
|
371 |
| "big M = 0" |
|
372 |
| "big N = 0" |
|
373 |
| "big P = 0" |
|
374 |
| "big Q = 0" |
|
375 |
| "big R = 0" |
|
376 |
| "big S = 0" |
|
377 |
| "big T = 0" |
|
378 |
| "big U = 0" |
|
379 |
| "big V = 0" |
|
380 |
||
381 |
||
382 |
(* automatic pattern splitting *) |
|
383 |
fun |
|
384 |
f4 :: "nat \<Rightarrow> nat \<Rightarrow> bool" |
|
385 |
where |
|
386 |
"f4 0 0 = True" |
|
25170 | 387 |
| "f4 _ _ = False" |
22726 | 388 |
|
19770
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HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
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diff
changeset
|
389 |
|
45008
8b74cfea913a
match types when applying mono_thm -- previous export generalizes type variables;
krauss
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diff
changeset
|
390 |
(* polymorphic partial_function *) |
8b74cfea913a
match types when applying mono_thm -- previous export generalizes type variables;
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changeset
|
391 |
partial_function (option) f5 :: "'a list \<Rightarrow> 'a option" |
8b74cfea913a
match types when applying mono_thm -- previous export generalizes type variables;
krauss
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41817
diff
changeset
|
392 |
where |
8b74cfea913a
match types when applying mono_thm -- previous export generalizes type variables;
krauss
parents:
41817
diff
changeset
|
393 |
"f5 x = f5 x" |
8b74cfea913a
match types when applying mono_thm -- previous export generalizes type variables;
krauss
parents:
41817
diff
changeset
|
394 |
|
19736 | 395 |
end |