--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/semiring_normalizer.ML Sat May 08 17:15:50 2010 +0200
@@ -0,0 +1,907 @@
+(* Title: HOL/Tools/Groebner_Basis/normalizer.ML
+ Author: Amine Chaieb, TU Muenchen
+
+Normalization of expressions in semirings.
+*)
+
+signature SEMIRING_NORMALIZER =
+sig
+ type entry
+ val get: Proof.context -> (thm * entry) list
+ val match: Proof.context -> cterm -> entry option
+ val del: attribute
+ val add: {semiring: cterm list * thm list, ring: cterm list * thm list,
+ field: cterm list * thm list, idom: thm list, ideal: thm list} -> attribute
+ val funs: thm -> {is_const: morphism -> cterm -> bool,
+ dest_const: morphism -> cterm -> Rat.rat,
+ mk_const: morphism -> ctyp -> Rat.rat -> cterm,
+ conv: morphism -> Proof.context -> cterm -> thm} -> declaration
+ val semiring_funs: thm -> declaration
+ val field_funs: thm -> declaration
+
+ 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: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
+ val semiring_normalizers_ord_wrapper: Proof.context -> entry ->
+ (cterm -> cterm -> bool) ->
+ {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
+
+ val setup: theory -> theory
+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;
+ val merge = AList.merge Thm.eq_thm (K true);
+);
+
+val get = Data.get o Context.Proof;
+
+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 as {is_const, dest_const, mk_const, conv}) 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 (get ctxt) end;
+
+
+(* logical content *)
+
+val semiringN = "semiring";
+val ringN = "ring";
+val idomN = "idom";
+val idealN = "ideal";
+val fieldN = "field";
+
+val del = Thm.declaration_attribute (Data.map o AList.delete Thm.eq_thm);
+
+fun add {semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules),
+ field = (f_ops, f_rules), idom, ideal} =
+ Thm.declaration_attribute (fn key => fn context => context |> Data.map
+ let
+ val ctxt = Context.proof_of context;
+
+ 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 37 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 |> pairself Thm.dest_binop;
+ val ((ca, cb), (cc, cd)) =
+ rule 20 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
+ val cm = rule 1 |> Thm.rhs_of |> Thm.dest_arg;
+ val (cp, cq) = rule 26 |> Thm.lhs_of |> Thm.dest_binop |> pairself 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 (symmetric o mk_meta) ideal
+ in
+ AList.delete Thm.eq_thm key #>
+ cons (key, ({vars = vars, semiring = semiring,
+ ring = ring, field = field, idom = idom, ideal = ideal'},
+ {is_const = undefined, dest_const = undefined, mk_const = undefined,
+ conv = undefined}))
+ end);
+
+
+(* extra-logical functions *)
+
+fun funs raw_key {is_const, dest_const, mk_const, conv} phi =
+ Data.map (fn data =>
+ let
+ val key = Morphism.thm phi raw_key;
+ val _ = AList.defined Thm.eq_thm data key orelse
+ raise THM ("No data entry for structure key", 0, [key]);
+ val fns = {is_const = is_const phi, dest_const = dest_const phi,
+ mk_const = mk_const phi, conv = conv phi};
+ in AList.map_entry Thm.eq_thm key (apsnd (K fns)) data end);
+
+fun semiring_funs key = funs key
+ {is_const = fn phi => can HOLogic.dest_number o Thm.term_of,
+ dest_const = fn phi => fn ct =>
+ Rat.rat_of_int (snd
+ (HOLogic.dest_number (Thm.term_of ct)
+ handle TERM _ => error "ring_dest_const")),
+ mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT
+ (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int"),
+ conv = fn phi => fn _ => Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm})
+ then_conv Simplifier.rewrite (HOL_basic_ss addsimps
+ (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}))};
+
+fun field_funs key =
+ let
+ fun numeral_is_const ct =
+ case 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 Rings.inverse},_)$t => can HOLogic.dest_number t
+ | t => can HOLogic.dest_number t
+ fun dest_const ct = ((case term_of ct of
+ Const (@{const_name Rings.divide},_) $ a $ b=>
+ Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | Const (@{const_name Rings.inverse},_)$t =>
+ Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
+ handle TERM _ => error "ring_dest_const")
+ fun mk_const phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Numeral.mk_cnumber cT a
+ else Thm.capply
+ (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+ (Numeral.mk_cnumber cT a))
+ (Numeral.mk_cnumber cT b)
+ end
+ in funs key
+ {is_const = K numeral_is_const,
+ dest_const = K dest_const,
+ mk_const = mk_const,
+ conv = K (K Numeral_Simprocs.field_comp_conv)}
+ 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 $ _ $ _ => 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}
+ @ [@{thm if_False}, @{thm 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 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,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_add_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 (Conv.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 = (Conv.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_exp_ss =
+ HOL_basic_ss addsimps (@{thms nat_number} @ @{thms nat_arith} @ @{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 (term_of t, 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 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 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;
+
+
+(** Isar setup **)
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
+fun keyword2 k1 k2 = Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.colon) >> K ();
+fun keyword3 k1 k2 k3 =
+ Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.$$$ k3 -- Args.colon) >> K ();
+
+val opsN = "ops";
+val rulesN = "rules";
+
+val normN = "norm";
+val constN = "const";
+val delN = "del";
+
+val any_keyword =
+ keyword2 semiringN opsN || keyword2 semiringN rulesN ||
+ keyword2 ringN opsN || keyword2 ringN rulesN ||
+ keyword2 fieldN opsN || keyword2 fieldN rulesN ||
+ keyword2 idomN rulesN || keyword2 idealN rulesN;
+
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+val terms = thms >> map Drule.dest_term;
+
+fun optional scan = Scan.optional scan [];
+
+in
+
+val setup =
+ Attrib.setup @{binding normalizer}
+ (Scan.lift (Args.$$$ delN >> K del) ||
+ ((keyword2 semiringN opsN |-- terms) --
+ (keyword2 semiringN rulesN |-- thms)) --
+ (optional (keyword2 ringN opsN |-- terms) --
+ optional (keyword2 ringN rulesN |-- thms)) --
+ (optional (keyword2 fieldN opsN |-- terms) --
+ optional (keyword2 fieldN rulesN |-- thms)) --
+ optional (keyword2 idomN rulesN |-- thms) --
+ optional (keyword2 idealN rulesN |-- thms)
+ >> (fn ((((sr, r), f), id), idl) =>
+ add {semiring = sr, ring = r, field = f, idom = id, ideal = idl}))
+ "semiring normalizer data";
+
+end;
+
+end;