# Crossing the River Safely¶

For the sake of brain scrambling, we're going to solve this ancient puzzle using Imandra (again!). As most polyvalent farmers will tell you, going to the market with your pet wolf, tastiest goat, and freshest cabbage is sometimes difficult as they tend to have appetite for one another. The good news is that there is a way to cross this river safely anyway.

First we should define the problem by tallying our goods and looking around.

In :
type location =
| Boat
| LeftCoast
| RightCoast
| Eaten

type boat =
| Left
| Right

type good = Cabbage | Goat | Wolf

Out:
type location = Boat | LeftCoast | RightCoast | Eaten
type boat = Left | Right
type good = Cabbage | Goat | Wolf


This problem is delicate and will require multiple steps to be solved. Each step should take us from a state to another state (where, hopefully, no cabbage nor goat was hurt).

In :
type state = {
cabbage : location;
goat : location;
wolf : location;
boat : boat;
}

Out:
type state = {
cabbage : location;
goat : location;
wolf : location;
boat : boat;
}


We can define a few helpers:

In :
let get_location (s:state) (g:good) = match g with
| Cabbage -> s.cabbage
| Goat -> s.goat
| Wolf -> s.wolf

let set_location (s:state) (g:good) (l:location) = match g with
| Cabbage -> { s with cabbage = l}
| Goat    -> { s with goat    = l}
| Wolf    -> { s with wolf    = l}

let boat_empty (s:state) =
(s.cabbage <> Boat) &&
(s.goat    <> Boat) &&
(s.wolf    <> Boat)

Out:
val get_location : state -> good -> location = <fun>
val set_location : state -> good -> location -> state = <fun>
val boat_empty : state -> bool = <fun>


Now, transition from a state to the next one is done via actions:

In :
type action =
| Pick of good
| Drop of good
| CrossRiver

let process_action (s:state) (m:action) : state =
match m with
| CrossRiver -> { s with boat = match s.boat with Left -> Right | Right -> Left }
| Pick x -> begin
if not @@ boat_empty s then s else
match get_location s x, s.boat with
|  LeftCoast ,  Left -> set_location s x Boat
| RightCoast , Right -> set_location s x Boat
| _ -> s
end
| Drop x -> begin
match get_location s x, s.boat with
| Boat ,  Left -> set_location s x LeftCoast
| Boat , Right -> set_location s x RightCoast
| _ -> s
end
;;

let process_eating s =
match s.boat, s.cabbage, s.goat, s.wolf with
| Right, LeftCoast, LeftCoast, _    -> Some { s with cabbage = Eaten }
| Right, _ , LeftCoast, LeftCoast   -> Some { s with goat = Eaten }
|  Left, RightCoast, RightCoast, _  -> Some { s with cabbage = Eaten }
|  Left, _ , RightCoast, RightCoast -> Some { s with goat = Eaten }
| _  -> None

(* is… it a bad state? *)
let anything_eaten s =
s.cabbage = Eaten || s.goat = Eaten

let one_step s a =
if anything_eaten s then s
else
match process_eating s with
| Some s -> s
| None ->
process_action s a

(* process many actions. Note that we have to specify that [acts] is
the argument that proves termination *)
let rec many_steps s acts =
match acts with
| [] -> s
| a :: acts ->
let s' = one_step s a in
many_steps s' acts

let solved s =
s.cabbage = RightCoast
&& s.goat = RightCoast
&& s.wolf = RightCoast
&& s.boat = Right
;;

Out:
type action = Pick of good | Drop of good | CrossRiver
val process_action : state -> action -> state = <fun>
val process_eating : state -> state option = <fun>
val anything_eaten : state -> bool = <fun>
val one_step : state -> action -> state = <fun>
val many_steps : state -> action list -> state = <fun>
val solved : state -> bool = <fun>

termination proof

### Termination proof

call many_steps (one_step s (List.hd acts)) (List.tl acts) from many_steps s acts
originalmany_steps s acts
submany_steps (one_step s (List.hd acts)) (List.tl acts)
original ordinalOrdinal.Int (Ordinal.count acts)
sub ordinalOrdinal.Int (Ordinal.count (List.tl acts))
path[not (acts = [])]
proof
detailed proof
ground_instances3
definitions0
inductions0
search_time
0.016s
details
Expand
smt_stats
 num checks 8 arith-make-feasible 21 arith-max-columns 16 arith-conflicts 1 rlimit count 2848 mk clause 24 datatype occurs check 36 mk bool var 120 arith-lower 13 datatype splits 22 decisions 40 propagations 27 arith-max-rows 6 conflicts 13 datatype accessor ax 15 datatype constructor ax 36 num allocs 8.34569e+08 final checks 9 added eqs 144 del clause 9 arith eq adapter 13 arith-upper 16 memory 19.66 max memory 19.66
Expand
• start[0.016s]
let (_x_0 : int) = Ordinal.count acts in
let (_x_1 : action list) = List.tl acts in
let (_x_2 : int) = Ordinal.count _x_1 in
not (acts = []) && _x_0 >= 0 && _x_2 >= 0
==> _x_1 = [] || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
• ##### simplify
 into let (_x_0 : action list) = List.tl acts in let (_x_1 : int) = Ordinal.count _x_0 in let (_x_2 : int) = Ordinal.count acts in (_x_0 = [] || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2)) || not ((not (acts = []) && _x_2 >= 0) && _x_1 >= 0) expansions [] rewrite_steps forward_chaining
• unroll
 expr (|count_action list_2492| acts_2482) expansions
• unroll
 expr (|count_action list_2492| (|get.::.1_2474| acts_2482)) expansions
• unroll
 expr (|Ordinal.<<_102| (|Ordinal.Int_93| (|count_action list_2492| … expansions
• Unsat

In :
(* initial state, on the west bank of Anduin with empty pockets and fuzzy side-kicks *)
let init_state = {
cabbage = LeftCoast;
goat = LeftCoast;
wolf = LeftCoast;
boat = Left;
}

Out:
val init_state : state =
{cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left}


We are now ready to ask for a solution! Because we're looking for a given solution rather than a universal proof, instance is the most natural.

In :
#timeout 10_000;;

instance (fun l -> solved @@ many_steps init_state l) ;;

Out:
- : action list -> bool = <fun>
module CX : sig val l : action list end

Instance (after 18 steps, 29.591s):
let (l : action list) =
[(Pick (Goat)); CrossRiver; (Drop (Goat)); CrossRiver; (Pick (Cabbage));
CrossRiver; (Drop (Cabbage)); (Pick (Goat)); CrossRiver; (Drop (Goat));
(Pick (Wolf)); CrossRiver; (Drop (Wolf)); CrossRiver; (Pick (Goat));
CrossRiver; (Drop (Goat))]

Instance
proof attempt
ground_instances18
definitions0
inductions0
search_time
29.591s
details
Expand
smt_stats
 num checks 37 arith-make-feasible 1 arith-max-columns 4 rlimit count 2.00718e+07 mk clause 47589 datatype occurs check 2916 restarts 348 mk bool var 155906 datatype splits 817412 decisions 185066 propagations 3.46076e+06 conflicts 46056 datatype accessor ax 8177 minimized lits 492018 datatype constructor ax 138478 final checks 70 added eqs 8.88891e+06 del clause 28820 dyn ack 31 memory 192.69 max memory 193.02 num allocs 2.71315e+10
Expand
• start[29.591s]
let (_x_0 : state)
= many_steps
{cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left}
:var_0:
in
_x_0.cabbage = RightCoast
&& _x_0.goat = RightCoast && _x_0.wolf = RightCoast && _x_0.boat = Right
• #### simplify

 into let (_x_0 : state) = many_steps {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} :var_0: in ((_x_0.cabbage = RightCoast && _x_0.goat = RightCoast) && _x_0.wolf = RightCoast) && _x_0.boat = Right expansions [] rewrite_steps forward_chaining
• unroll
 expr (many_steps_60 (|rec_mk.state_2526| LeftCoast_16 LeftCoast_16 … expansions
• unroll
 expr (many_steps_60 (one_step_55 (|rec_mk.state_2526| LeftCoast_16 … expansions
• unroll
 expr (many_steps_60 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• unroll
 expr (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526| … expansions
• Sat (Some let (l : action list) = [(Pick (Goat)); CrossRiver; (Drop (Goat)); CrossRiver; (Pick (Cabbage)); CrossRiver; (Drop (Cabbage)); (Pick (Goat)); CrossRiver; (Drop (Goat)); (Pick (Wolf)); CrossRiver; (Drop (Wolf)); CrossRiver; (Pick (Goat)); CrossRiver; (Drop (Goat))] )

That seems to take a bit of time, because this problem is not that easy for Imandra's unrolling algorithm. Let's try [@@blast] to see if we can get a result faster:

In :
instance (fun l -> solved @@ many_steps init_state l)
[@@blast] ;;

Out:
- : action list -> bool = <fun>
module CX : sig val l : action list end

Instance (after 72 steps, 0.171s):
let l =
let (_x_0 : action) = Pick Goat in
let (_x_1 : action) = Drop Goat in
[_x_0; CrossRiver; _x_1; CrossRiver; Pick Cabbage; CrossRiver;
Drop Cabbage; _x_0; CrossRiver; _x_1; Pick Wolf; CrossRiver; Drop Wolf;
CrossRiver; _x_0; CrossRiver; _x_1]

Instance

It only took a fraction of second! 🎉

Now we have a clear plan for crossing the river. How to sell the goat and cabbage is left as an exercise to the reader.