    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.SBV.Examples.CodeGeneration.GCD
    -- Copyright   :  (c) Levent Erkok
    -- License     :  BSD3
    -- Maintainer  :  erkokl@gmail.com
    -- Stability   :  experimental
    --
    -- Computing GCD symbolically, and generating C code for it. This example
    -- illustrates symbolic termination related issues when programming with
    -- SBV, when the termination of a recursive algorithm crucially depends
    -- on the value of a symbolic variable. The technique we use is to statically
    -- enforce termination by using a recursion depth counter.
    -----------------------------------------------------------------------------

    module Data.SBV.Examples.CodeGeneration.GCD where

    import Data.SBV

    -----------------------------------------------------------------------------
    -- * Computing GCD
    -----------------------------------------------------------------------------

    -- | The symbolic GCD algorithm, over two 8-bit numbers. We define @sgcd a 0@ to
    -- be @a@ for all @a@, which implies @sgcd 0 0 = 0@. Note that this is essentially
    -- Euclid's algorithm, except with a recursion depth counter. We need the depth
    -- counter since the algorithm is not /symbolically terminating/, as we don't have
    -- a means of determining that the second argument (@b@) will eventually reach 0 in a symbolic
    -- context. Hence we stop after 12 iterations. Why 12? We've empirically determined that this
    -- algorithm will recurse at most 12 times for arbitrary 8-bit numbers. Of course, this is
    -- a claim that we shall prove below.
    sgcd :: SWord8 -> SWord8 -> SWord8
    sgcd a b = go a b 12
      where go :: SWord8 -> SWord8 -> SWord8 -> SWord8
            go x y c = ite (c .== 0 ||| y .== 0)   -- stop if y is 0, or if we reach the recursion depth
                           x
                           (go y y' (c-1))
              where (_, y') = x `sQuotRem` y

    -----------------------------------------------------------------------------
    -- * Verification
    -----------------------------------------------------------------------------

    {- $VerificationIntro
    We prove that 'sgcd' does indeed compute the common divisor of the given numbers.
    Our predicate takes @x@, @y@, and @k@. We show that what 'sgcd' returns is indeed a common divisor,
    and it is at least as large as any given @k@, provided @k@ is a common divisor as well.
    -}

    -- | We have:
    --
    -- >>> prove sgcdIsCorrect
    -- Q.E.D.
    sgcdIsCorrect :: SWord8 -> SWord8 -> SWord8 -> SBool
    sgcdIsCorrect x y k = ite (y  .== 0)                        -- if y is 0
                              (k' .== x)                        -- then k' must be x, nothing else to prove by definition
                              (isCommonDivisor k'  &&&          -- otherwise, k' is a common divisor and
                              (isCommonDivisor k ==> k' .>= k)) -- if k is a common divisor as well, then k' is at least as large as k
      where k' = sgcd x y
            isCommonDivisor a = z1 .== 0 &&& z2 .== 0
               where (_, z1) = x `sQuotRem` a
                     (_, z2) = y `sQuotRem` a

    -----------------------------------------------------------------------------
    -- * Code generation
    -----------------------------------------------------------------------------

    {- $VerificationIntro
    Now that we have proof our 'sgcd' implementation is correct, we can go ahead
    and generate C code for it.
    -}

    -- | This call will generate the required C files. The following is the function
    -- body generated for 'sgcd'. (We are not showing the generated header, @Makefile@,
    -- and the driver programs for brevity.) Note that the generated function is
    -- a constant time algorithm for GCD. It is not necessarily fastest, but it will take
    -- precisely the same amount of time for all values of @x@ and @y@.
    --
    -- > /* File: "sgcd.c". Automatically generated by SBV. Do not edit! */
    -- >
    -- > #include <stdio.h>
    -- > #include <stdlib.h>
    -- > #include <inttypes.h>
    -- > #include <stdint.h>
    -- > #include <stdbool.h>
    -- > #include "sgcd.h"
    -- >
    -- > SWord8 sgcd(const SWord8 x, const SWord8 y)
    -- > {
    -- >   const SWord8 s0 = x;
    -- >   const SWord8 s1 = y;
    -- >   const SBool  s3 = s1 == 0;
    -- >   const SWord8 s4 = (s1 == 0) ? s0 : (s0 % s1);
    -- >   const SWord8 s5 = s3 ? s0 : s4;
    -- >   const SBool  s6 = 0 == s5;
    -- >   const SWord8 s7 = (s5 == 0) ? s1 : (s1 % s5);
    -- >   const SWord8 s8 = s6 ? s1 : s7;
    -- >   const SBool  s9 = 0 == s8;
    -- >   const SWord8 s10 = (s8 == 0) ? s5 : (s5 % s8);
    -- >   const SWord8 s11 = s9 ? s5 : s10;
    -- >   const SBool  s12 = 0 == s11;
    -- >   const SWord8 s13 = (s11 == 0) ? s8 : (s8 % s11);
    -- >   const SWord8 s14 = s12 ? s8 : s13;
    -- >   const SBool  s15 = 0 == s14;
    -- >   const SWord8 s16 = (s14 == 0) ? s11 : (s11 % s14);
    -- >   const SWord8 s17 = s15 ? s11 : s16;
    -- >   const SBool  s18 = 0 == s17;
    -- >   const SWord8 s19 = (s17 == 0) ? s14 : (s14 % s17);
    -- >   const SWord8 s20 = s18 ? s14 : s19;
    -- >   const SBool  s21 = 0 == s20;
    -- >   const SWord8 s22 = (s20 == 0) ? s17 : (s17 % s20);
    -- >   const SWord8 s23 = s21 ? s17 : s22;
    -- >   const SBool  s24 = 0 == s23;
    -- >   const SWord8 s25 = (s23 == 0) ? s20 : (s20 % s23);
    -- >   const SWord8 s26 = s24 ? s20 : s25;
    -- >   const SBool  s27 = 0 == s26;
    -- >   const SWord8 s28 = (s26 == 0) ? s23 : (s23 % s26);
    -- >   const SWord8 s29 = s27 ? s23 : s28;
    -- >   const SBool  s30 = 0 == s29;
    -- >   const SWord8 s31 = (s29 == 0) ? s26 : (s26 % s29);
    -- >   const SWord8 s32 = s30 ? s26 : s31;
    -- >   const SBool  s33 = 0 == s32;
    -- >   const SWord8 s34 = (s32 == 0) ? s29 : (s29 % s32);
    -- >   const SWord8 s35 = s33 ? s29 : s34;
    -- >   const SBool  s36 = 0 == s35;
    -- >   const SWord8 s37 = s36 ? s32 : s35;
    -- >   const SWord8 s38 = s33 ? s29 : s37;
    -- >   const SWord8 s39 = s30 ? s26 : s38;
    -- >   const SWord8 s40 = s27 ? s23 : s39;
    -- >   const SWord8 s41 = s24 ? s20 : s40;
    -- >   const SWord8 s42 = s21 ? s17 : s41;
    -- >   const SWord8 s43 = s18 ? s14 : s42;
    -- >   const SWord8 s44 = s15 ? s11 : s43;
    -- >   const SWord8 s45 = s12 ? s8 : s44;
    -- >   const SWord8 s46 = s9 ? s5 : s45;
    -- >   const SWord8 s47 = s6 ? s1 : s46;
    -- >   const SWord8 s48 = s3 ? s0 : s47;
    -- >
    -- >   return s48;
    -- > }
    genGCDInC :: IO ()
    genGCDInC = compileToC Nothing "sgcd" $ do
                    x <- cgInput "x"
                    y <- cgInput "y"
                    cgReturn $ sgcd x y