(* ID: $Id$
Author: Amine Chaieb
Tactic for solving equalities over commutative rings.
*)
signature COMM_RING =
sig
val comm_ring_tac : int -> tactic
val comm_ring_method: int -> Proof.method
val algebra_method: int -> Proof.method
val setup : (theory -> theory) list
end
structure CommRing: COMM_RING =
struct
(* The Cring exception for erronous uses of cring_tac *)
exception CRing of string;
(* Zero and One of the commutative ring *)
fun cring_zero T = Const("0",T);
fun cring_one T = Const("1",T);
(* reification functions *)
(* add two polynom expressions *)
fun polT t = Type ("Commutative_Ring.pol",[t]);
fun polexT t = Type("Commutative_Ring.polex",[t]);
val nT = HOLogic.natT;
fun listT T = Type ("List.list",[T]);
(* Reification of the constructors *)
(* Nat*)
val succ = Const("Suc",nT --> nT);
val zero = Const("0",nT);
val one = Const("1",nT);
(* Lists *)
fun reif_list T [] = Const("List.list.Nil",listT T)
| reif_list T (x::xs) = Const("List.list.Cons",[T,listT T] ---> listT T)
$x$(reif_list T xs);
(* pol*)
fun pol_Pc t = Const("Commutative_Ring.pol.Pc",t --> polT t);
fun pol_Pinj t = Const("Commutative_Ring.pol.Pinj",[nT,polT t] ---> polT t);
fun pol_PX t = Const("Commutative_Ring.pol.PX",[polT t, nT, polT t] ---> polT t);
(* polex *)
fun polex_add t = Const("Commutative_Ring.polex.Add",[polexT t,polexT t] ---> polexT t);
fun polex_sub t = Const("Commutative_Ring.polex.Sub",[polexT t,polexT t] ---> polexT t);
fun polex_mul t = Const("Commutative_Ring.polex.Mul",[polexT t,polexT t] ---> polexT t);
fun polex_neg t = Const("Commutative_Ring.polex.Neg",polexT t --> polexT t);
fun polex_pol t = Const("Commutative_Ring.polex.Pol",polT t --> polexT t);
fun polex_pow t = Const("Commutative_Ring.polex.Pow",[polexT t, nT] ---> polexT t);
(* reification of natural numbers *)
fun reif_nat n =
if n>0 then succ$(reif_nat (n-1))
else if n=0 then zero
else raise CRing "ring_tac: reif_nat negative n";
(* reification of polynoms : primitive cring expressions *)
fun reif_pol T vs t =
case t of
Free(_,_) =>
let val i = find_index_eq t vs
in if i = 0
then (pol_PX T)$((pol_Pc T)$ (cring_one T))
$one$((pol_Pc T)$(cring_zero T))
else (pol_Pinj T)$(reif_nat i)$
((pol_PX T)$((pol_Pc T)$ (cring_one T))
$one$
((pol_Pc T)$(cring_zero T)))
end
| _ => (pol_Pc T)$ t;
(* reification of polynom expressions *)
fun reif_polex T vs t =
case t of
Const("op +",_)$a$b => (polex_add T)
$ (reif_polex T vs a)$(reif_polex T vs b)
| Const("op -",_)$a$b => (polex_sub T)
$ (reif_polex T vs a)$(reif_polex T vs b)
| Const("op *",_)$a$b => (polex_mul T)
$ (reif_polex T vs a)$ (reif_polex T vs b)
| Const("uminus",_)$a => (polex_neg T)
$ (reif_polex T vs a)
| (Const("Nat.power",_)$a$n) => (polex_pow T) $ (reif_polex T vs a) $ n
| _ => (polex_pol T) $ (reif_pol T vs t);
(* reification of the equation *)
val cr_sort = Sign.read_sort (the_context ()) "{comm_ring,recpower}";
fun reif_eq sg (eq as Const("op =",Type("fun",a::_))$lhs$rhs) =
if Sign.of_sort (the_context()) (a,cr_sort)
then
let val fs = term_frees eq
val cvs = cterm_of sg (reif_list a fs)
val clhs = cterm_of sg (reif_polex a fs lhs)
val crhs = cterm_of sg (reif_polex a fs rhs)
val ca = ctyp_of sg a
in (ca,cvs,clhs, crhs)
end
else raise CRing "reif_eq: not an equation over comm_ring + recpower"
| reif_eq sg _ = raise CRing "reif_eq: not an equation";
(*The cring tactic *)
(* Attention: You have to make sure that no t^0 is in the goal!! *)
(* Use simply rewriting t^0 = 1 *)
fun cring_ss sg = simpset_of sg
addsimps
(map thm ["mkPX_def", "mkPinj_def","sub_def",
"power_add","even_def","pow_if"])
addsimps [sym OF [thm "power_add"]];
val norm_eq = thm "norm_eq"
fun comm_ring_tac i =(fn st =>
let
val g = List.nth (prems_of st, i - 1)
val sg = sign_of_thm st
val (ca,cvs,clhs,crhs) = reif_eq sg (HOLogic.dest_Trueprop g)
val norm_eq_th = simplify (cring_ss sg)
(instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs]
norm_eq)
in ((cut_rules_tac [norm_eq_th] i)
THEN (simp_tac (cring_ss sg) i)
THEN (simp_tac (cring_ss sg) i)) st
end);
fun comm_ring_method i = Method.METHOD (fn facts =>
Method.insert_tac facts 1 THEN comm_ring_tac i);
val algebra_method = comm_ring_method;
val setup =
[Method.add_method ("comm_ring",
Method.no_args (comm_ring_method 1),
"reflective decision procedure for equalities over commutative rings"),
Method.add_method ("algebra",
Method.no_args (algebra_method 1),
"Method for proving algebraic properties: for now only comm_ring")];
end;