# 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
original:many_steps s acts
sub:many_steps (one_step s (List.hd acts)) (List.tl acts)
original ordinal:Ordinal.Int (_cnt acts)
sub ordinal:Ordinal.Int (_cnt (List.tl acts))
path:[acts <> []]
proof:
detailed proof
ground_instances:3
definitions:0
inductions:0
search_time:
0.013s
details:
Expand
smt_stats:
 num checks: 8 arith assert lower: 14 arith tableau max rows: 5 arith tableau max columns: 15 arith pivots: 8 rlimit count: 2723 mk clause: 29 datatype occurs check: 22 mk bool var: 119 arith assert upper: 12 datatype splits: 14 decisions: 36 arith row summations: 9 propagations: 30 conflicts: 13 arith fixed eqs: 5 datatype accessor ax: 17 arith conflicts: 2 arith num rows: 5 arith assert diseq: 1 datatype constructor ax: 33 num allocs: 7.00661e+06 final checks: 6 added eqs: 117 del clause: 12 arith eq adapter: 13 memory: 17.37 max memory: 17.37
Expand
• start[0.013s]
let (_x_0 : int) = count.list count.action acts in
let (_x_1 : action list) = List.tl acts in
let (_x_2 : int) = count.list count.action _x_1 in
acts <> [] && ((_x_0 >= 0) && (_x_2 >= 0))
==> not (_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) = count.list count.action _x_0 in let (_x_2 : int) = count.list count.action acts in not (_x_0 <> []) || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2) || not (acts <> [] && (_x_2 >= 0) && (_x_1 >= 0)) expansions: [] rewrite_steps: forward_chaining:
• unroll
 expr: (|Ordinal.<<| (|Ordinal.Int_79/boot| (|count.list_436/server| (|ge… expansions:
• unroll
 expr: (|count.list_436/server| (|get.::.1_416/server| acts_424/server)) expansions:
• unroll
 expr: (|count.list_436/server| acts_424/server) 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, 35.905s):
let l : action list =
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
proof attempt
ground_instances:18
definitions:0
inductions:0
search_time:
35.905s
details:
Expand
smt_stats:
 num checks: 37 rlimit count: 4.47397e+07 mk clause: 81926 datatype occurs check: 1178 restarts: 507 mk bool var: 1.13465e+06 datatype splits: 1.94881e+06 decisions: 301257 propagations: 8.06683e+06 conflicts: 80381 datatype accessor ax: 15597 minimized lits: 921501 datatype constructor ax: 1.35596e+06 final checks: 36 added eqs: 2.05255e+07 del clause: 61434 dyn ack: 42 time: 4.754 memory: 348.68 max memory: 349.06 num allocs: 1.5752e+11
Expand
• start[35.905s]
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_1322/client (|rec_mk.state_473/server| LeftCoast_1253/client LeftCoast_1253/cl… expansions:
• unroll
 expr: (many_steps_1322/client (one_step_1318/client (|rec_mk.state_473/server| LeftCoast_1253/… expansions:
• unroll
 expr: (many_steps_1322/client (one_step_1318/client (one_step_1318/client (|rec_mk.state_473/s… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• unroll
 expr: (let ((a!1 (one_step_1318/client (one_step_1318/client (one_step_1318/cl… expansions:
• Sat (Some let l : action list = 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] )

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, 0.112s):
let l : action list =
let (_x_0 : action) = Pick Goat in
let (_x_1 : action) = Drop Goat in
[_x_0; CrossRiver; _x_1; CrossRiver; Pick Wolf; CrossRiver; Drop Wolf;
_x_0; CrossRiver; _x_1; Pick Cabbage; CrossRiver; Drop Cabbage;
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.