(* Title: HOL/Tools/Groebner_Basis/normalizer.ML
Author: Amine Chaieb, TU Muenchen
*)
signature NORMALIZER =
sig
val semiring_normalize_conv : Proof.context -> conv
val semiring_normalize_ord_conv : Proof.context -> (cterm -> cterm -> bool) -> conv
val semiring_normalize_tac : Proof.context -> int -> tactic
val semiring_normalize_wrapper : Proof.context -> NormalizerData.entry -> conv
val semiring_normalizers_ord_wrapper :
Proof.context -> NormalizerData.entry -> (cterm -> cterm -> bool) ->
{add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
val semiring_normalize_ord_wrapper : Proof.context -> NormalizerData.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: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
end
structure Normalizer: NORMALIZER =
struct
open Conv;
(* Very basic stuff for terms *)
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 $ _ $ _ => 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 ([], inst);
val dest_numeral = term_of #> HOLogic.dest_number #> snd;
val is_numeral = can dest_numeral;
val numeral01_conv = Simplifier.rewrite
(HOL_basic_ss addsimps [@{thm numeral_1_eq_1}, @{thm numeral_0_eq_0}]);
val zero1_numeral_conv =
Simplifier.rewrite (HOL_basic_ss addsimps [@{thm numeral_1_eq_1} RS sym, @{thm numeral_0_eq_0} RS sym]);
fun zerone_conv 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 @{thms arith_simps} @ natarith @ @{thms rel_simps}
@ [if_False, if_True, @{thm Nat.add_0}, @{thm add_Suc},
@{thm add_number_of_left}, @{thm Suc_eq_plus1}]
@ map (fn th => th RS sym) @{thms 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) (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,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
([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
| _ => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), K false, true_tm, true_tm));
val (divide_inverse, inverse_divide, divide_tm, inverse_tm, is_divide) =
(case (f_ops, f_rules) of
([divide_pat, inverse_pat], [div_inv, inv_div]) =>
let val div_tm = funpow 2 Thm.dest_fun divide_pat
val inv_tm = Thm.dest_fun inverse_pat
in (div_inv, inv_div, div_tm, inv_tm, is_binop div_tm)
end
| _ => (TrueI, TrueI, true_tm, true_tm, K false));
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 (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. *)
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). *)
fun num_conv n =
nat_add_conv (Thm.capply @{cterm Suc} (Numeral.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. *)
fun polynomial_neg_conv 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. *)
fun polynomial_sub_conv 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 is_semiring_constant tm then semiring_add_conv tm
else if not(is_comb 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 lopr aconvc inverse_tm then
let val th1 = Drule.arg_cong_rule lopr (polynomial_conv r)
in transitive th1 (semiring_mul_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 divide_tm
then
let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l))
(polynomial_conv r)
val th2 = (rewr_conv divide_inverse then_conv polynomial_mul_conv)
(Thm.rhs_of th1)
in transitive th1 th2
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 (@{thms nat_number} @ nat_arith @ @{thms arith_simps} @ @{thms rel_simps})
addsimps [Let_def, if_False, if_True, @{thm Nat.add_0}, @{thm add_Suc}];
fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS;
fun semiring_normalizers_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal},
{conv, dest_const, mk_const, is_const}) ord =
let
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 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);
fun semiring_normalize_wrapper ctxt data =
semiring_normalize_ord_wrapper ctxt data simple_cterm_ord;
fun semiring_normalize_ord_conv ctxt ord tm =
(case NormalizerData.match ctxt tm of
NONE => 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;
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;