```-----------------------------------------------------------------------------
-- |
-- Module    : Documentation.SBV.Examples.ProofTools.Sum
-- Copyright : (c) Levent Erkok
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Example inductive proof to show partial correctness of the traditional
-- for-loop sum algorithm:
--
-- @
--     s = 0
--     i = 0
--     while i <= n:
--        s += i
--        i++
-- @
--
-- We prove the loop invariant and establish partial correctness that
-- @s@ is the sum of all numbers up to and including @n@ upon termination.
-----------------------------------------------------------------------------

{-# LANGUAGE DeriveAnyClass        #-}
{-# LANGUAGE DeriveFoldable        #-}
{-# LANGUAGE DeriveGeneric         #-}
{-# LANGUAGE DeriveTraversable     #-}
{-# LANGUAGE FlexibleInstances     #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE NamedFieldPuns        #-}

module Documentation.SBV.Examples.ProofTools.Sum where

import Data.SBV
import Data.SBV.Tools.Induction
import Data.SBV.Control

import GHC.Generics hiding (S)

-- * System state

-- | System state. We simply have two components, parameterized
-- over the type so we can put in both concrete and symbolic values.
data S a = S { s :: a, i :: a, n :: a } deriving (Show, Mergeable, Generic, Functor, Foldable, Traversable)

-- | 'Fresh' instance for our state
instance Fresh IO (S SInteger) where
fresh  = S <\$> freshVar_  <*> freshVar_  <*> freshVar_

-- | Encoding partial correctness of the sum algorithm. We have:
--
-- >>> sumCorrect
-- Q.E.D.
sumCorrect :: IO (InductionResult (S Integer))
sumCorrect = induct chatty setup initial trans strengthenings inv goal
where -- Set this to True for SBV to print steps as it proceeds
-- through the inductive proof
chatty :: Bool
chatty = False

-- This is where we would put solver options, typically via
-- calls to 'Data.SBV.setOption'. We do not need any for this problem,
-- so we simply do nothing.
setup :: Symbolic ()
setup = return ()

-- Initially, @s@ and @i@ are both @0@. We also require @n@ to be at least @0@.
initial :: S SInteger -> SBool
initial S{s, i, n} = s .== 0 .&& i .== 0 .&& n .>= 0

-- We code the algorithm almost literally in SBV notation:
trans :: S SInteger -> [S SInteger]
trans st@S{s, i, n} = [ite (i .<= n)
st { s = s+i, i = i+1 }
st
]

-- No strengthenings needed for this problem!
strengthenings :: [(String, S SInteger -> SBool)]
strengthenings = []

-- Loop invariant: @i@ remains at most @n+1@ and @s@ the sum of
-- all the numbers up-to @i-1@.
inv :: S SInteger -> SBool
inv S{s, i, n} =    i .<= n+1
.&& s .== (i * (i - 1)) `sDiv` 2

-- Final goal. When the termination condition holds, the sum is
-- equal to all the numbers up to and including @n@. Note that
-- SBV does not prove the termination condition; it simply is
-- the indication that the loop has ended as specified by the user.
goal :: S SInteger -> (SBool, SBool)
goal S{s, i, n} = (i .== n+1, s .== (n * (n+1)) `sDiv` 2)
```