(* Title: HOL/Tools/semiring_normalizer.ML
Author: Amine Chaieb, TU Muenchen
Normalization of expressions in semirings.
*)
signature SEMIRING_NORMALIZER =
sig
type entry
val match: Proof.context -> cterm -> entry option
val the_semiring: Proof.context -> thm -> cterm list * thm list
val the_ring: Proof.context -> thm -> cterm list * thm list
val the_field: Proof.context -> thm -> cterm list * thm list
val the_idom: Proof.context -> thm -> thm list
val the_ideal: Proof.context -> thm -> thm list
val declare: thm -> {semiring: term list * thm list, ring: term list * thm list,
field: term list * thm list, idom: thm list, ideal: thm list} ->
local_theory -> local_theory
val semiring_normalize_conv: Proof.context -> conv
val semiring_normalize_ord_conv: Proof.context -> (cterm -> cterm -> bool) -> conv
val semiring_normalize_wrapper: Proof.context -> entry -> conv
val semiring_normalize_ord_wrapper: Proof.context -> entry
-> (cterm -> cterm -> bool) -> conv
val semiring_normalizers_conv: cterm list -> cterm list * thm list
-> cterm list * thm list -> cterm list * thm list ->
(cterm -> bool) * conv * conv * conv -> (cterm -> cterm -> bool) ->
{add: Proof.context -> conv,
mul: Proof.context -> conv,
neg: Proof.context -> conv,
main: Proof.context -> conv,
pow: Proof.context -> conv,
sub: Proof.context -> conv}
val semiring_normalizers_ord_wrapper: Proof.context -> entry ->
(cterm -> cterm -> bool) ->
{add: Proof.context -> conv,
mul: Proof.context -> conv,
neg: Proof.context -> conv,
main: Proof.context -> conv,
pow: Proof.context -> conv,
sub: Proof.context -> conv}
end
structure Semiring_Normalizer: SEMIRING_NORMALIZER =
struct
(** data **)
type entry =
{vars: cterm list,
semiring: cterm list * thm list,
ring: cterm list * thm list,
field: cterm list * thm list,
idom: thm list,
ideal: thm list} *
{is_const: cterm -> bool,
dest_const: cterm -> Rat.rat,
mk_const: ctyp -> Rat.rat -> cterm,
conv: Proof.context -> cterm -> thm};
structure Data = Generic_Data
(
type T = (thm * entry) list;
val empty = [];
val extend = I;
fun merge data = AList.merge Thm.eq_thm (K true) data;
);
fun the_rules ctxt = fst o the o AList.lookup Thm.eq_thm (Data.get (Context.Proof ctxt))
val the_semiring = #semiring oo the_rules
val the_ring = #ring oo the_rules
val the_field = #field oo the_rules
val the_idom = #idom oo the_rules
val the_ideal = #ideal oo the_rules
fun match ctxt tm =
let
fun match_inst
({vars, semiring = (sr_ops, sr_rules),
ring = (r_ops, r_rules), field = (f_ops, f_rules), idom, ideal},
fns) pat =
let
fun h instT =
let
val substT = Thm.instantiate (instT, []);
val substT_cterm = Drule.cterm_rule substT;
val vars' = map substT_cterm vars;
val semiring' = (map substT_cterm sr_ops, map substT sr_rules);
val ring' = (map substT_cterm r_ops, map substT r_rules);
val field' = (map substT_cterm f_ops, map substT f_rules);
val idom' = map substT idom;
val ideal' = map substT ideal;
val result = ({vars = vars', semiring = semiring',
ring = ring', field = field', idom = idom', ideal = ideal'}, fns);
in SOME result end
in (case try Thm.match (pat, tm) of
NONE => NONE
| SOME (instT, _) => h instT)
end;
fun match_struct (_,
entry as ({semiring = (sr_ops, _), ring = (r_ops, _), field = (f_ops, _), ...}, _): entry) =
get_first (match_inst entry) (sr_ops @ r_ops @ f_ops);
in get_first match_struct (Data.get (Context.Proof ctxt)) end;
(* extra-logical functions *)
val semiring_norm_ss =
simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms semiring_norm});
val semiring_funs =
{is_const = can HOLogic.dest_number o Thm.term_of,
dest_const = (fn ct =>
Rat.of_int (snd
(HOLogic.dest_number (Thm.term_of ct)
handle TERM _ => error "ring_dest_const"))),
mk_const = (fn cT => fn x => Numeral.mk_cnumber cT
(case Rat.dest x of (i, 1) => i | _ => error "int_of_rat: bad int")),
conv = (fn ctxt =>
Simplifier.rewrite (put_simpset semiring_norm_ss ctxt)
then_conv Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms numeral_One}))};
val divide_const = Thm.cterm_of @{context} (Logic.varify_global @{term "op /"});
val [divide_tvar] = Term.add_tvars (Thm.term_of divide_const) [];
val field_funs =
let
fun numeral_is_const ct =
case Thm.term_of ct of
Const (@{const_name Rings.divide},_) $ a $ b =>
can HOLogic.dest_number a andalso can HOLogic.dest_number b
| Const (@{const_name Fields.inverse},_)$t => can HOLogic.dest_number t
| t => can HOLogic.dest_number t
fun dest_const ct = ((case Thm.term_of ct of
Const (@{const_name Rings.divide},_) $ a $ b=>
Rat.make (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
| Const (@{const_name Fields.inverse},_)$t =>
Rat.inv (Rat.of_int (snd (HOLogic.dest_number t)))
| t => Rat.of_int (snd (HOLogic.dest_number t)))
handle TERM _ => error "ring_dest_const")
fun mk_const cT x =
let val (a, b) = Rat.dest x
in if b = 1 then Numeral.mk_cnumber cT a
else Thm.apply
(Thm.apply (Thm.instantiate_cterm ([(divide_tvar, cT)], []) divide_const)
(Numeral.mk_cnumber cT a))
(Numeral.mk_cnumber cT b)
end
in
{is_const = numeral_is_const,
dest_const = dest_const,
mk_const = mk_const,
conv = Numeral_Simprocs.field_comp_conv}
end;
(* logical content *)
val semiringN = "semiring";
val ringN = "ring";
val fieldN = "field";
val idomN = "idom";
fun declare raw_key
{semiring = raw_semiring0, ring = raw_ring0, field = raw_field0, idom = raw_idom, ideal = raw_ideal}
lthy =
let
val ctxt' = fold Variable.auto_fixes (fst raw_semiring0 @ fst raw_ring0 @ fst raw_field0) lthy;
val prepare_ops = apfst (Variable.export_terms ctxt' lthy #> map (Thm.cterm_of lthy));
val raw_semiring = prepare_ops raw_semiring0;
val raw_ring = prepare_ops raw_ring0;
val raw_field = prepare_ops raw_field0;
in
lthy |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi => fn context =>
let
val ctxt = Context.proof_of context;
val key = Morphism.thm phi raw_key;
fun transform_ops_rules (ops, rules) =
(map (Morphism.cterm phi) ops, Morphism.fact phi rules);
val (sr_ops, sr_rules) = transform_ops_rules raw_semiring;
val (r_ops, r_rules) = transform_ops_rules raw_ring;
val (f_ops, f_rules) = transform_ops_rules raw_field;
val idom = Morphism.fact phi raw_idom;
val ideal = Morphism.fact phi raw_ideal;
fun check kind name xs n =
null xs orelse length xs = n orelse
error ("Expected " ^ string_of_int n ^ " " ^ kind ^ " for " ^ name);
val check_ops = check "operations";
val check_rules = check "rules";
val _ =
check_ops semiringN sr_ops 5 andalso
check_rules semiringN sr_rules 36 andalso
check_ops ringN r_ops 2 andalso
check_rules ringN r_rules 2 andalso
check_ops fieldN f_ops 2 andalso
check_rules fieldN f_rules 2 andalso
check_rules idomN idom 2;
val mk_meta = Local_Defs.meta_rewrite_rule ctxt;
val sr_rules' = map mk_meta sr_rules;
val r_rules' = map mk_meta r_rules;
val f_rules' = map mk_meta f_rules;
fun rule i = nth sr_rules' (i - 1);
val (cx, cy) = Thm.dest_binop (hd sr_ops);
val cz = rule 34 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
val cn = rule 36 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
val ((clx, crx), (cly, cry)) =
rule 13 |> Thm.rhs_of |> Thm.dest_binop |> apply2 Thm.dest_binop;
val ((ca, cb), (cc, cd)) =
rule 20 |> Thm.lhs_of |> Thm.dest_binop |> apply2 Thm.dest_binop;
val cm = rule 1 |> Thm.rhs_of |> Thm.dest_arg;
val (cp, cq) = rule 26 |> Thm.lhs_of |> Thm.dest_binop |> apply2 Thm.dest_arg;
val vars = [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry];
val semiring = (sr_ops, sr_rules');
val ring = (r_ops, r_rules');
val field = (f_ops, f_rules');
val ideal' = map (Thm.symmetric o mk_meta) ideal
in
context
|> Data.map (AList.update Thm.eq_thm (key,
({vars = vars, semiring = semiring, ring = ring, field = field, idom = idom, ideal = ideal'},
(if null f_ops then semiring_funs else field_funs))))
end)
end;
(** auxiliary **)
fun is_comb ct =
(case Thm.term_of ct of
_ $ _ => true
| _ => false);
val concl = Thm.cprop_of #> Thm.dest_arg;
fun is_binop ct ct' =
(case Thm.term_of ct' of
c $ _ $ _ => Thm.term_of ct aconv c
| _ => false);
fun dest_binop ct ct' =
if is_binop ct ct' then Thm.dest_binop ct'
else raise CTERM ("dest_binop: bad binop", [ct, ct'])
fun inst_thm inst = Thm.instantiate ([], map (apfst (dest_Var o Thm.term_of)) inst);
val dest_number = Thm.term_of #> HOLogic.dest_number #> snd;
val is_number = can dest_number;
fun numeral01_conv ctxt =
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_One}]);
fun zero1_numeral_conv ctxt =
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_One} RS sym]);
fun zerone_conv ctxt cv =
zero1_numeral_conv ctxt then_conv cv then_conv numeral01_conv ctxt;
val nat_add_ss = simpset_of
(put_simpset HOL_basic_ss @{context}
addsimps @{thms arith_simps} @ @{thms diff_nat_numeral} @ @{thms rel_simps}
@ @{thms if_False if_True Nat.add_0 add_Suc add_numeral_left Suc_eq_plus1}
@ map (fn th => th RS sym) @{thms numerals});
fun nat_add_conv ctxt =
zerone_conv ctxt (Simplifier.rewrite (put_simpset nat_add_ss ctxt));
val zeron_tm = @{cterm "0::nat"};
val onen_tm = @{cterm "1::nat"};
val true_tm = @{cterm "True"};
(** normalizing conversions **)
(* core conversion *)
fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_rules) (f_ops, f_rules)
(is_semiring_constant, semiring_add_conv, semiring_mul_conv, semiring_pow_conv) =
let
val [pthm_02, pthm_03, pthm_04, pthm_05, pthm_07, pthm_08,
pthm_09, pthm_10, pthm_11, pthm_12, pthm_13, pthm_14, pthm_15, pthm_16,
pthm_17, pthm_18, pthm_19, pthm_21, pthm_22, pthm_23, pthm_24,
pthm_25, pthm_26, pthm_27, pthm_28, pthm_29, pthm_30, pthm_31, pthm_32,
pthm_33, pthm_34, pthm_35, pthm_36, pthm_37, pthm_38, _] = sr_rules;
val [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry] = vars;
val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
val [add_tm, mul_tm, pow_tm] = map (Thm.dest_fun o Thm.dest_fun) [add_pat, mul_pat, pow_pat];
val dest_add = dest_binop add_tm
val dest_mul = dest_binop mul_tm
fun dest_pow tm =
let val (l,r) = dest_binop pow_tm tm
in if is_number r then (l,r) else raise CTERM ("dest_pow",[tm])
end;
val is_add = is_binop add_tm
val is_mul = is_binop mul_tm
val (neg_mul, sub_add, sub_tm, neg_tm, dest_sub, cx', cy') =
(case (r_ops, r_rules) of
([sub_pat, neg_pat], [neg_mul, sub_add]) =>
let
val sub_tm = Thm.dest_fun (Thm.dest_fun sub_pat)
val neg_tm = Thm.dest_fun neg_pat
val dest_sub = dest_binop sub_tm
in (neg_mul, sub_add, sub_tm, neg_tm, dest_sub, neg_mul |> concl |> Thm.dest_arg,
sub_add |> concl |> Thm.dest_arg |> Thm.dest_arg)
end
| _ => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), true_tm, true_tm));
val (divide_inverse, divide_tm, inverse_tm) =
(case (f_ops, f_rules) of
([divide_pat, inverse_pat], [div_inv, _]) =>
let val div_tm = funpow 2 Thm.dest_fun divide_pat
val inv_tm = Thm.dest_fun inverse_pat
in (div_inv, div_tm, inv_tm)
end
| _ => (TrueI, true_tm, true_tm));
in fn variable_order =>
let
(* Conversion for "x^n * x^m", with either x^n = x and/or x^m = x possible. *)
(* Also deals with "const * const", but both terms must involve powers of *)
(* the same variable, or both be constants, or behaviour may be incorrect. *)
fun powvar_mul_conv ctxt tm =
let
val (l,r) = dest_mul tm
in if is_semiring_constant l andalso is_semiring_constant r
then semiring_mul_conv tm
else
((let
val (lx,ln) = dest_pow l
in
((let val (_, rn) = dest_pow r
val th1 = inst_thm [(cx,lx),(cp,ln),(cq,rn)] pthm_29
val (tm1,tm2) = Thm.dest_comb(concl th1) in
Thm.transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv ctxt tm2)) end)
handle CTERM _ =>
(let val th1 = inst_thm [(cx,lx),(cq,ln)] pthm_31
val (tm1,tm2) = Thm.dest_comb(concl th1) in
Thm.transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv ctxt tm2)) end)) end)
handle CTERM _ =>
((let val (rx,rn) = dest_pow r
val th1 = inst_thm [(cx,rx),(cq,rn)] pthm_30
val (tm1,tm2) = Thm.dest_comb(concl th1) in
Thm.transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv ctxt tm2)) end)
handle CTERM _ => inst_thm [(cx,l)] pthm_32
))
end;
(* Remove "1 * m" from a monomial, and just leave m. *)
fun monomial_deone th =
(let val (l,r) = dest_mul(concl th) in
if l aconvc one_tm
then Thm.transitive th (inst_thm [(ca,r)] pthm_13) else th end)
handle CTERM _ => th;
(* Conversion for "(monomial)^n", where n is a numeral. *)
fun monomial_pow_conv ctxt =
let
fun monomial_pow tm bod ntm =
if not(is_comb bod)
then Thm.reflexive tm
else
if is_semiring_constant bod
then semiring_pow_conv tm
else
let
val (lopr,r) = Thm.dest_comb bod
in if not(is_comb lopr)
then Thm.reflexive tm
else
let
val (opr,l) = Thm.dest_comb lopr
in
if opr aconvc pow_tm andalso is_number r
then
let val th1 = inst_thm [(cx,l),(cp,r),(cq,ntm)] pthm_34
val (l,r) = Thm.dest_comb(concl th1)
in Thm.transitive th1 (Drule.arg_cong_rule l (nat_add_conv ctxt r))
end
else
if opr aconvc mul_tm
then
let
val th1 = inst_thm [(cx,l),(cy,r),(cq,ntm)] pthm_33
val (xy,z) = Thm.dest_comb(concl th1)
val (x,y) = Thm.dest_comb xy
val thl = monomial_pow y l ntm
val thr = monomial_pow z r ntm
in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule x thl) thr)
end
else Thm.reflexive tm
end
end
in fn tm =>
let
val (lopr,r) = Thm.dest_comb tm
val (opr,l) = Thm.dest_comb lopr
in if not (opr aconvc pow_tm) orelse not(is_number r)
then raise CTERM ("monomial_pow_conv", [tm])
else if r aconvc zeron_tm
then inst_thm [(cx,l)] pthm_35
else if r aconvc onen_tm
then inst_thm [(cx,l)] pthm_36
else monomial_deone(monomial_pow tm l r)
end
end;
(* Multiplication of canonical monomials. *)
fun monomial_mul_conv ctxt =
let
fun powvar tm =
if is_semiring_constant tm then one_tm
else
((let val (lopr,r) = Thm.dest_comb tm
val (opr,l) = Thm.dest_comb lopr
in if opr aconvc pow_tm andalso is_number r then l
else raise CTERM ("monomial_mul_conv",[tm]) end)
handle CTERM _ => tm) (* FIXME !? *)
fun vorder x y =
if x aconvc y then 0
else
if x aconvc one_tm then ~1
else if y aconvc one_tm then 1
else if variable_order x y then ~1 else 1
fun monomial_mul tm l r =
((let val (lx,ly) = dest_mul l val vl = powvar lx
in
((let
val (rx,ry) = dest_mul r
val vr = powvar rx
val ord = vorder vl vr
in
if ord = 0
then
let
val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] pthm_15
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv ctxt tm4)) tm2
val th3 = Thm.transitive th1 th2
val (tm5,tm6) = Thm.dest_comb(concl th3)
val (tm7,tm8) = Thm.dest_comb tm6
val th4 = monomial_mul tm6 (Thm.dest_arg tm7) tm8
in Thm.transitive th3 (Drule.arg_cong_rule tm5 th4)
end
else
let val th0 = if ord < 0 then pthm_16 else pthm_17
val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] th0
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm2
in Thm.transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
end
end)
handle CTERM _ =>
(let val vr = powvar r val ord = vorder vl vr
in
if ord = 0 then
let
val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_18
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv ctxt tm4)) tm2
in Thm.transitive th1 th2
end
else
if ord < 0 then
let val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_19
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm2
in Thm.transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
end
else inst_thm [(ca,l),(cb,r)] pthm_09
end)) end)
handle CTERM _ =>
(let val vl = powvar l in
((let
val (rx,ry) = dest_mul r
val vr = powvar rx
val ord = vorder vl vr
in if ord = 0 then
let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_21
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
in Thm.transitive th1 (Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv ctxt tm4)) tm2)
end
else if ord > 0 then
let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_22
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm2
in Thm.transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
end
else Thm.reflexive tm
end)
handle CTERM _ =>
(let val vr = powvar r
val ord = vorder vl vr
in if ord = 0 then powvar_mul_conv ctxt tm
else if ord > 0 then inst_thm [(ca,l),(cb,r)] pthm_09
else Thm.reflexive tm
end)) end))
in fn tm => let val (l,r) = dest_mul tm in monomial_deone(monomial_mul tm l r)
end
end;
(* Multiplication by monomial of a polynomial. *)
fun polynomial_monomial_mul_conv ctxt =
let
fun pmm_conv tm =
let val (l,r) = dest_mul tm
in
((let val (y,z) = dest_add r
val th1 = inst_thm [(cx,l),(cy,y),(cz,z)] pthm_37
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 =
Thm.combination (Drule.arg_cong_rule tm3 (monomial_mul_conv ctxt tm4)) (pmm_conv tm2)
in Thm.transitive th1 th2
end)
handle CTERM _ => monomial_mul_conv ctxt tm)
end
in pmm_conv
end;
(* Addition of two monomials identical except for constant multiples. *)
fun monomial_add_conv tm =
let val (l,r) = dest_add tm
in if is_semiring_constant l andalso is_semiring_constant r
then semiring_add_conv tm
else
let val th1 =
if is_mul l andalso is_semiring_constant(Thm.dest_arg1 l)
then if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r) then
inst_thm [(ca,Thm.dest_arg1 l),(cm,Thm.dest_arg r), (cb,Thm.dest_arg1 r)] pthm_02
else inst_thm [(ca,Thm.dest_arg1 l),(cm,r)] pthm_03
else if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r)
then inst_thm [(cm,l),(ca,Thm.dest_arg1 r)] pthm_04
else inst_thm [(cm,r)] pthm_05
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.arg_cong_rule tm3 (semiring_add_conv tm4)
val th3 = Thm.transitive th1 (Drule.fun_cong_rule th2 tm2)
val tm5 = concl th3
in
if (Thm.dest_arg1 tm5) aconvc zero_tm
then Thm.transitive th3 (inst_thm [(ca,Thm.dest_arg tm5)] pthm_11)
else monomial_deone th3
end
end;
(* Ordering on monomials. *)
fun striplist dest =
let fun strip x acc =
((let val (l,r) = dest x in
strip l (strip r acc) end)
handle CTERM _ => x::acc) (* FIXME !? *)
in fn x => strip x []
end;
fun powervars tm =
let val ptms = striplist dest_mul tm
in if is_semiring_constant (hd ptms) then tl ptms else ptms
end;
val num_0 = 0;
val num_1 = 1;
fun dest_varpow tm =
((let val (x,n) = dest_pow tm in (x,dest_number n) end)
handle CTERM _ =>
(tm,(if is_semiring_constant tm then num_0 else num_1)));
val morder =
let fun lexorder l1 l2 =
case (l1,l2) of
([],[]) => 0
| (_ ,[]) => ~1
| ([], _) => 1
| (((x1,n1)::vs1),((x2,n2)::vs2)) =>
if variable_order x1 x2 then 1
else if variable_order x2 x1 then ~1
else if n1 < n2 then ~1
else if n2 < n1 then 1
else lexorder vs1 vs2
in fn tm1 => fn tm2 =>
let val vdegs1 = map dest_varpow (powervars tm1)
val vdegs2 = map dest_varpow (powervars tm2)
val deg1 = fold (Integer.add o snd) vdegs1 num_0
val deg2 = fold (Integer.add o snd) vdegs2 num_0
in if deg1 < deg2 then ~1 else if deg1 > deg2 then 1
else lexorder vdegs1 vdegs2
end
end;
(* Addition of two polynomials. *)
fun polynomial_add_conv ctxt =
let
fun dezero_rule th =
let
val tm = concl th
in
if not(is_add tm) then th else
let val (lopr,r) = Thm.dest_comb tm
val l = Thm.dest_arg lopr
in
if l aconvc zero_tm
then Thm.transitive th (inst_thm [(ca,r)] pthm_07) else
if r aconvc zero_tm
then Thm.transitive th (inst_thm [(ca,l)] pthm_08) else th
end
end
fun padd tm =
let
val (l,r) = dest_add tm
in
if l aconvc zero_tm then inst_thm [(ca,r)] pthm_07
else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_08
else
if is_add l
then
let val (a,b) = dest_add l
in
if is_add r then
let val (c,d) = dest_add r
val ord = morder a c
in
if ord = 0 then
let val th1 = inst_thm [(ca,a),(cb,b),(cc,c),(cd,d)] pthm_23
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.arg_cong_rule tm3 (monomial_add_conv tm4)
in dezero_rule (Thm.transitive th1 (Thm.combination th2 (padd tm2)))
end
else (* ord <> 0*)
let val th1 =
if ord > 0 then inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
else inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
val (tm1,tm2) = Thm.dest_comb(concl th1)
in dezero_rule (Thm.transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
end
end
else (* not (is_add r)*)
let val ord = morder a r
in
if ord = 0 then
let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_26
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
in dezero_rule (Thm.transitive th1 th2)
end
else (* ord <> 0*)
if ord > 0 then
let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
val (tm1,tm2) = Thm.dest_comb(concl th1)
in dezero_rule (Thm.transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
end
else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
end
end
else (* not (is_add l)*)
if is_add r then
let val (c,d) = dest_add r
val ord = morder l c
in
if ord = 0 then
let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_28
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
in dezero_rule (Thm.transitive th1 th2)
end
else
if ord > 0 then Thm.reflexive tm
else
let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
val (tm1,tm2) = Thm.dest_comb(concl th1)
in dezero_rule (Thm.transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
end
end
else
let val ord = morder l r
in
if ord = 0 then monomial_add_conv tm
else if ord > 0 then dezero_rule(Thm.reflexive tm)
else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
end
end
in padd
end;
(* Multiplication of two polynomials. *)
fun polynomial_mul_conv ctxt =
let
fun pmul tm =
let val (l,r) = dest_mul tm
in
if not(is_add l) then polynomial_monomial_mul_conv ctxt tm
else
if not(is_add r) then
let val th1 = inst_thm [(ca,l),(cb,r)] pthm_09
in Thm.transitive th1 (polynomial_monomial_mul_conv ctxt (concl th1))
end
else
let val (a,b) = dest_add l
val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_10
val (tm1,tm2) = Thm.dest_comb(concl th1)
val (tm3,tm4) = Thm.dest_comb tm1
val th2 = Drule.arg_cong_rule tm3 (polynomial_monomial_mul_conv ctxt tm4)
val th3 = Thm.transitive th1 (Thm.combination th2 (pmul tm2))
in Thm.transitive th3 (polynomial_add_conv ctxt (concl th3))
end
end
in fn tm =>
let val (l,r) = dest_mul tm
in
if l aconvc zero_tm then inst_thm [(ca,r)] pthm_11
else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_12
else if l aconvc one_tm then inst_thm [(ca,r)] pthm_13
else if r aconvc one_tm then inst_thm [(ca,l)] pthm_14
else pmul tm
end
end;
(* Power of polynomial (optimized for the monomial and trivial cases). *)
fun num_conv ctxt n =
nat_add_conv ctxt (Thm.apply @{cterm Suc} (Numeral.mk_cnumber @{ctyp nat} (dest_number n - 1)))
|> Thm.symmetric;
fun polynomial_pow_conv ctxt =
let
fun ppow tm =
let val (l,n) = dest_pow tm
in
if n aconvc zeron_tm then inst_thm [(cx,l)] pthm_35
else if n aconvc onen_tm then inst_thm [(cx,l)] pthm_36
else
let val th1 = num_conv ctxt n
val th2 = inst_thm [(cx,l),(cq,Thm.dest_arg (concl th1))] pthm_38
val (tm1,tm2) = Thm.dest_comb(concl th2)
val th3 = Thm.transitive th2 (Drule.arg_cong_rule tm1 (ppow tm2))
val th4 = Thm.transitive (Drule.arg_cong_rule (Thm.dest_fun tm) th1) th3
in Thm.transitive th4 (polynomial_mul_conv ctxt (concl th4))
end
end
in fn tm =>
if is_add(Thm.dest_arg1 tm) then ppow tm else monomial_pow_conv ctxt tm
end;
(* Negation. *)
fun polynomial_neg_conv ctxt tm =
let val (l,r) = Thm.dest_comb tm in
if not (l aconvc neg_tm) then raise CTERM ("polynomial_neg_conv",[tm]) else
let val th1 = inst_thm [(cx', r)] neg_mul
val th2 = Thm.transitive th1 (Conv.arg1_conv semiring_mul_conv (concl th1))
in Thm.transitive th2 (polynomial_monomial_mul_conv ctxt (concl th2))
end
end;
(* Subtraction. *)
fun polynomial_sub_conv ctxt tm =
let val (l,r) = dest_sub tm
val th1 = inst_thm [(cx', l), (cy', r)] sub_add
val (tm1,tm2) = Thm.dest_comb(concl th1)
val th2 = Drule.arg_cong_rule tm1 (polynomial_neg_conv ctxt tm2)
in Thm.transitive th1 (Thm.transitive th2 (polynomial_add_conv ctxt (concl th2)))
end;
(* Conversion from HOL term. *)
fun polynomial_conv ctxt tm =
if is_semiring_constant tm then semiring_add_conv tm
else if not(is_comb tm) then Thm.reflexive tm
else
let val (lopr,r) = Thm.dest_comb tm
in if lopr aconvc neg_tm then
let val th1 = Drule.arg_cong_rule lopr (polynomial_conv ctxt r)
in Thm.transitive th1 (polynomial_neg_conv ctxt (concl th1))
end
else if lopr aconvc inverse_tm then
let val th1 = Drule.arg_cong_rule lopr (polynomial_conv ctxt r)
in Thm.transitive th1 (semiring_mul_conv (concl th1))
end
else
if not(is_comb lopr) then Thm.reflexive tm
else
let val (opr,l) = Thm.dest_comb lopr
in if opr aconvc pow_tm andalso is_number r
then
let val th1 = Drule.fun_cong_rule (Drule.arg_cong_rule opr (polynomial_conv ctxt l)) r
in Thm.transitive th1 (polynomial_pow_conv ctxt (concl th1))
end
else if opr aconvc divide_tm
then
let val th1 = Thm.combination (Drule.arg_cong_rule opr (polynomial_conv ctxt l))
(polynomial_conv ctxt r)
val th2 = (Conv.rewr_conv divide_inverse then_conv polynomial_mul_conv ctxt)
(Thm.rhs_of th1)
in Thm.transitive th1 th2
end
else
if opr aconvc add_tm orelse opr aconvc mul_tm orelse opr aconvc sub_tm
then
let val th1 =
Thm.combination
(Drule.arg_cong_rule opr (polynomial_conv ctxt l)) (polynomial_conv ctxt r)
val f = if opr aconvc add_tm then polynomial_add_conv ctxt
else if opr aconvc mul_tm then polynomial_mul_conv ctxt
else polynomial_sub_conv ctxt
in Thm.transitive th1 (f (concl th1))
end
else Thm.reflexive tm
end
end;
in
{main = polynomial_conv,
add = polynomial_add_conv,
mul = polynomial_mul_conv,
pow = polynomial_pow_conv,
neg = polynomial_neg_conv,
sub = polynomial_sub_conv}
end
end;
val nat_exp_ss =
simpset_of
(put_simpset HOL_basic_ss @{context}
addsimps (@{thms eval_nat_numeral} @ @{thms diff_nat_numeral} @ @{thms arith_simps} @ @{thms rel_simps})
addsimps [@{thm Let_def}, @{thm if_False}, @{thm if_True}, @{thm Nat.add_0}, @{thm add_Suc}]);
fun simple_cterm_ord t u = Term_Ord.term_ord (Thm.term_of t, Thm.term_of u) = LESS;
(* various normalizing conversions *)
fun semiring_normalizers_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal},
{conv, dest_const, mk_const, is_const}) ord =
let
val pow_conv =
Conv.arg_conv (Simplifier.rewrite (put_simpset nat_exp_ss ctxt))
then_conv Simplifier.rewrite
(put_simpset HOL_basic_ss ctxt addsimps [nth (snd semiring) 31, nth (snd semiring) 34])
then_conv conv ctxt
val dat = (is_const, conv ctxt, conv ctxt, pow_conv)
in semiring_normalizers_conv vars semiring ring field dat ord end;
fun semiring_normalize_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal}, {conv, dest_const, mk_const, is_const}) ord =
#main (semiring_normalizers_ord_wrapper ctxt
({vars = vars, semiring = semiring, ring = ring, field = field, idom = idom, ideal = ideal},
{conv = conv, dest_const = dest_const, mk_const = mk_const, is_const = is_const}) ord) ctxt;
fun semiring_normalize_wrapper ctxt data =
semiring_normalize_ord_wrapper ctxt data simple_cterm_ord;
fun semiring_normalize_ord_conv ctxt ord tm =
(case match ctxt tm of
NONE => Thm.reflexive tm
| SOME res => semiring_normalize_ord_wrapper ctxt res ord tm);
fun semiring_normalize_conv ctxt = semiring_normalize_ord_conv ctxt simple_cterm_ord;
end;