--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Groebner_Basis/normalizer.ML Tue Jun 05 16:26:04 2007 +0200
@@ -0,0 +1,649 @@
+(* Title: HOL/Tools/Groebner_Basis/normalizer.ML
+ ID: $Id$
+ Author: Amine Chaieb, TU Muenchen
+*)
+
+signature NORMALIZER =
+sig
+ val mk_cnumber : ctyp -> int -> cterm
+ val mk_cnumeral : int -> cterm
+ val semiring_normalize_conv : Proof.context -> Conv.conv
+ val semiring_normalize_tac : Proof.context -> int -> tactic
+ val semiring_normalize_wrapper : NormalizerData.entry -> Conv.conv
+ val semiring_normalizers_conv :
+ cterm list -> cterm list * thm list -> cterm list * thm list ->
+ (cterm -> bool) * Conv.conv * Conv.conv * Conv.conv -> (cterm -> Thm.cterm -> bool) ->
+ {add: Conv.conv, mul: Conv.conv, neg: Conv.conv, main: Conv.conv,
+ pow: Conv.conv, sub: Conv.conv}
+end
+
+structure Normalizer: NORMALIZER =
+struct
+open Misc;
+
+local
+ val pls_const = @{cterm "Numeral.Pls"}
+ and min_const = @{cterm "Numeral.Min"}
+ and bit_const = @{cterm "Numeral.Bit"}
+ and zero = @{cpat "0"}
+ and one = @{cpat "1"}
+ fun mk_cbit 0 = @{cterm "Numeral.bit.B0"}
+ | mk_cbit 1 = @{cterm "Numeral.bit.B1"}
+ | mk_cbit _ = raise CTERM ("mk_cbit", []);
+
+in
+
+fun mk_cnumeral 0 = pls_const
+ | mk_cnumeral ~1 = min_const
+ | mk_cnumeral i =
+ let val (q, r) = IntInf.divMod (i, 2)
+ in Thm.capply (Thm.capply bit_const (mk_cnumeral q)) (mk_cbit (IntInf.toInt r))
+ end;
+
+fun mk_cnumber cT =
+ let
+ val [nb_of, z, on] =
+ map (Drule.cterm_rule (instantiate' [SOME cT] [])) [@{cpat "number_of"}, zero, one]
+ fun h 0 = z
+ | h 1 = on
+ | h x = Thm.capply nb_of (mk_cnumeral x)
+ in h end;
+end;
+
+
+(* Very basic stuff for terms *)
+val dest_numeral = term_of #> HOLogic.dest_number #> snd;
+val is_numeral = can dest_numeral;
+
+val numeral01_conv = Simplifier.rewrite
+ (HOL_basic_ss addsimps [numeral_1_eq_1, numeral_0_eq_0]);
+val zero1_numeral_conv =
+ Simplifier.rewrite (HOL_basic_ss addsimps [numeral_1_eq_1 RS sym, numeral_0_eq_0 RS sym]);
+val zerone_conv = fn cv => zero1_numeral_conv then_conv cv then_conv numeral01_conv;
+val natarith = [@{thm "add_nat_number_of"}, @{thm "diff_nat_number_of"},
+ @{thm "mult_nat_number_of"}, @{thm "eq_nat_number_of"},
+ @{thm "less_nat_number_of"}];
+val nat_add_conv =
+ zerone_conv
+ (Simplifier.rewrite
+ (HOL_basic_ss
+ addsimps arith_simps @ natarith @ rel_simps
+ @ [if_False, if_True, add_0, add_Suc, add_number_of_left, Suc_eq_add_numeral_1]
+ @ map (fn th => th RS sym) numerals));
+
+val nat_mul_conv = nat_add_conv;
+val zeron_tm = @{cterm "0::nat"};
+val onen_tm = @{cterm "1::nat"};
+val true_tm = @{cterm "True"};
+
+
+(* The main function! *)
+fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_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,pthm_39,pthm_40] = 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_numeral 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
+fun is_pow tm = is_binop pow_tm tm andalso is_numeral(Thm.dest_arg tm);
+
+val (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub,cx',cy') =
+ (case (r_ops, r_rules) of
+ ([], []) => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), K false, true_tm, true_tm)
+ | ([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
+ val is_sub = is_binop sub_tm
+ in (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub, neg_mul |> concl |> Thm.dest_arg,
+ sub_add |> concl |> Thm.dest_arg |> Thm.dest_arg)
+ end);
+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 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 (rx,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
+ transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)
+ handle CTERM _ =>
+ (let val th1 = inst_thm [(cx,lx),(cq,ln)] pthm_31
+ val (tm1,tm2) = Thm.dest_comb(concl th1) in
+ transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv 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
+ transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv 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 transitive th (inst_thm [(ca,r)] pthm_13) else th end)
+ handle CTERM _ => th;
+
+(* Conversion for "(monomial)^n", where n is a numeral. *)
+
+ val monomial_pow_conv =
+ let
+ fun monomial_pow tm bod ntm =
+ if not(is_comb bod)
+ then 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 reflexive tm
+ else
+ let
+ val (opr,l) = Thm.dest_comb lopr
+ in
+ if opr aconvc pow_tm andalso is_numeral r
+ then
+ let val th1 = inst_thm [(cx,l),(cp,r),(cq,ntm)] pthm_34
+ val (l,r) = Thm.dest_comb(concl th1)
+ in transitive th1 (Drule.arg_cong_rule l (nat_mul_conv 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 transitive th1 (combination (Drule.arg_cong_rule x thl) thr)
+ end
+ else 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_numeral 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. *)
+ val monomial_mul_conv =
+ 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_numeral 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 tm4)) tm2
+ val th3 = 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 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 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 tm4)) tm2
+ in 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 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 transitive th1 (Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv 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 transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
+ end
+ else reflexive tm
+ end)
+ handle CTERM _ =>
+ (let val vr = powvar r
+ val ord = vorder vl vr
+ in if ord = 0 then powvar_mul_conv tm
+ else if ord > 0 then inst_thm [(ca,l),(cb,r)] pthm_09
+ else 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. *)
+
+ val polynomial_monomial_mul_conv =
+ 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 = combination (Drule.arg_cong_rule tm3 (monomial_mul_conv tm4)) (pmm_conv tm2)
+ in transitive th1 th2
+ end)
+ handle CTERM _ => monomial_mul_conv 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 = transitive th1 (Drule.fun_cong_rule th2 tm2)
+ val tm5 = concl th3
+ in
+ if (Thm.dest_arg1 tm5) aconvc zero_tm
+ then 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_numeral 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
+ | (vps,[]) => ~1
+ | ([],vps) => 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_rev ((curry (op +)) o snd) vdegs1 num_0
+ val deg2 = fold_rev ((curry (op +)) 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. *)
+
+val polynomial_add_conv =
+ 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 transitive th (inst_thm [(ca,r)] pthm_07) else
+ if r aconvc zero_tm
+ then 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 (transitive th1 (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 (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 (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 (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 (transitive th1 th2)
+ end
+ else
+ if ord > 0 then 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 (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(reflexive tm)
+ else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
+ end
+ end
+ in padd
+ end;
+
+(* Multiplication of two polynomials. *)
+
+val polynomial_mul_conv =
+ let
+ fun pmul tm =
+ let val (l,r) = dest_mul tm
+ in
+ if not(is_add l) then polynomial_monomial_mul_conv tm
+ else
+ if not(is_add r) then
+ let val th1 = inst_thm [(ca,l),(cb,r)] pthm_09
+ in transitive th1 (polynomial_monomial_mul_conv(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 tm4)
+ val th3 = transitive th1 (combination th2 (pmul tm2))
+ in transitive th3 (polynomial_add_conv (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). *)
+
+val Succ = @{cterm "Suc"};
+val num_conv = fn n =>
+ nat_add_conv (Thm.capply (Succ) (mk_cnumber @{ctyp "nat"} ((dest_numeral n) - 1)))
+ |> Thm.symmetric;
+
+
+val polynomial_pow_conv =
+ 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 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 = transitive th2 (Drule.arg_cong_rule tm1 (ppow tm2))
+ val th4 = transitive (Drule.arg_cong_rule (Thm.dest_fun tm) th1) th3
+ in transitive th4 (polynomial_mul_conv (concl th4))
+ end
+ end
+ in fn tm =>
+ if is_add(Thm.dest_arg1 tm) then ppow tm else monomial_pow_conv tm
+ end;
+
+(* Negation. *)
+
+val polynomial_neg_conv =
+ fn 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 = transitive th1 (arg1_conv semiring_mul_conv (concl th1))
+ in transitive th2 (polynomial_monomial_mul_conv (concl th2))
+ end
+ end;
+
+
+(* Subtraction. *)
+val polynomial_sub_conv = fn 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 tm2)
+ in transitive th1 (transitive th2 (polynomial_add_conv (concl th2)))
+ end;
+
+(* Conversion from HOL term. *)
+
+fun polynomial_conv tm =
+ if not(is_comb tm) orelse is_semiring_constant tm
+ then 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 r)
+ in transitive th1 (polynomial_neg_conv (concl th1))
+ end
+ else
+ if not(is_comb lopr) then reflexive tm
+ else
+ let val (opr,l) = Thm.dest_comb lopr
+ in if opr aconvc pow_tm andalso is_numeral r
+ then
+ let val th1 = Drule.fun_cong_rule (Drule.arg_cong_rule opr (polynomial_conv l)) r
+ in transitive th1 (polynomial_pow_conv (concl th1))
+ end
+ else
+ if opr aconvc add_tm orelse opr aconvc mul_tm orelse opr aconvc sub_tm
+ then
+ let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l)) (polynomial_conv r)
+ val f = if opr aconvc add_tm then polynomial_add_conv
+ else if opr aconvc mul_tm then polynomial_mul_conv
+ else polynomial_sub_conv
+ in transitive th1 (f (concl th1))
+ end
+ else 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_arith = @{thms "nat_arith"};
+val nat_exp_ss = HOL_basic_ss addsimps (nat_number @ nat_arith @ arith_simps @ rel_simps)
+ addsimps [Let_def, if_False, if_True, add_0, add_Suc];
+
+fun semiring_normalize_wrapper ({vars, semiring, ring, idom},
+ {conv, dest_const, mk_const, is_const}) =
+ let
+ fun ord t u = Term.term_ord (term_of t, term_of u) = LESS
+
+ val pow_conv =
+ arg_conv (Simplifier.rewrite nat_exp_ss)
+ then_conv Simplifier.rewrite
+ (HOL_basic_ss addsimps [nth (snd semiring) 31, nth (snd semiring) 34])
+ then_conv conv
+ val dat = (is_const, conv, conv, pow_conv)
+ val {main, ...} = semiring_normalizers_conv vars semiring ring dat ord
+ in main end;
+
+fun semiring_normalize_conv ctxt tm =
+ (case NormalizerData.match ctxt tm of
+ NONE => reflexive tm
+ | SOME res => semiring_normalize_wrapper res tm);
+
+
+fun semiring_normalize_tac ctxt = SUBGOAL (fn (goal, i) =>
+ rtac (semiring_normalize_conv ctxt
+ (cterm_of (ProofContext.theory_of ctxt) (fst (Logic.dest_equals goal)))) i);
+end;