wenzelm@33443: (*  Title:      HOL/Library/positivstellensatz.ML
wenzelm@33443:     Author:     Amine Chaieb, University of Cambridge
wenzelm@33443: 
wenzelm@33443: A generic arithmetic prover based on Positivstellensatz certificates
wenzelm@33443: --- also implements Fourrier-Motzkin elimination as a special case
wenzelm@33443: Fourrier-Motzkin elimination.
chaieb@31120: *)
chaieb@31120: 
chaieb@31120: (* A functor for finite mappings based on Tables *)
Philipp@32645: 
huffman@46594: signature FUNC =
chaieb@31120: sig
huffman@46594:   include TABLE
huffman@46594:   val apply : 'a table -> key -> 'a
huffman@46594:   val applyd :'a table -> (key -> 'a) -> key -> 'a
huffman@46594:   val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
huffman@46594:   val dom : 'a table -> key list
huffman@46594:   val tryapplyd : 'a table -> key -> 'a -> 'a
huffman@46594:   val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
huffman@46594:   val choose : 'a table -> key * 'a
huffman@46594:   val onefunc : key * 'a -> 'a table
chaieb@31120: end;
chaieb@31120: 
huffman@46594: functor FuncFun(Key: KEY) : FUNC =
chaieb@31120: struct
chaieb@31120: 
wenzelm@31971: structure Tab = Table(Key);
chaieb@31120: 
Philipp@32829: open Tab;
Philipp@32829: 
chaieb@31120: fun dom a = sort Key.ord (Tab.keys a);
huffman@46594: fun applyd f d x = case Tab.lookup f x of
chaieb@31120:    SOME y => y
chaieb@31120:  | NONE => d x;
chaieb@31120: 
chaieb@31120: fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
chaieb@31120: fun tryapplyd f a d = applyd f (K d) a;
chaieb@31120: fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
huffman@46594: fun combine f z a b =
huffman@46594:   let
huffman@46594:     fun h (k,v) t = case Tab.lookup t k of
huffman@46594:         NONE => Tab.update (k,v) t
huffman@46594:       | SOME v' => let val w = f v v'
huffman@46594:         in if z w then Tab.delete k t else Tab.update (k,w) t end;
chaieb@31120:   in Tab.fold h a b end;
chaieb@31120: 
huffman@46594: fun choose f = case Tab.min_key f of
huffman@46594:     SOME k => (k, the (Tab.lookup f k))
huffman@46594:   | NONE => error "FuncFun.choose : Completely empty function"
chaieb@31120: 
Philipp@32829: fun onefunc kv = update kv empty
Philipp@32829: 
chaieb@31120: end;
chaieb@31120: 
Philipp@32645: (* Some standard functors and utility functions for them *)
Philipp@32645: 
Philipp@32645: structure FuncUtil =
Philipp@32645: struct
Philipp@32645: 
chaieb@31120: structure Intfunc = FuncFun(type key = int val ord = int_ord);
Philipp@32645: structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
Philipp@32645: structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
chaieb@31120: structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
wenzelm@35408: structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
Philipp@32645: 
wenzelm@35408: val cterm_ord = Term_Ord.fast_term_ord o pairself term_of
Philipp@32645: 
Philipp@32645: structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
Philipp@32645: 
Philipp@32829: type monomial = int Ctermfunc.table;
chaieb@31120: 
Philipp@32829: val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
Philipp@32645: 
Philipp@32645: structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
chaieb@31120: 
Philipp@32829: type poly = Rat.rat Monomialfunc.table;
Philipp@32645: 
Philipp@32645: (* The ordering so we can create canonical HOL polynomials.                  *)
chaieb@31120: 
Philipp@32829: fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
chaieb@31120: 
Philipp@32645: fun monomial_order (m1,m2) =
huffman@46594:   if Ctermfunc.is_empty m2 then LESS
huffman@46594:   else if Ctermfunc.is_empty m1 then GREATER
huffman@46594:   else
huffman@46594:     let
huffman@46594:       val mon1 = dest_monomial m1
Philipp@32645:       val mon2 = dest_monomial m2
wenzelm@33002:       val deg1 = fold (Integer.add o snd) mon1 0
huffman@46594:       val deg2 = fold (Integer.add o snd) mon2 0
huffman@46594:     in if deg1 < deg2 then GREATER
huffman@46594:        else if deg1 > deg2 then LESS
huffman@46594:        else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
huffman@46594:     end;
chaieb@31120: 
Philipp@32645: end
chaieb@31120: 
Philipp@32645: (* positivstellensatz datatype and prover generation *)
chaieb@31120: 
huffman@46594: signature REAL_ARITH =
chaieb@31120: sig
huffman@46594: 
chaieb@31120:   datatype positivstellensatz =
huffman@46594:     Axiom_eq of int
huffman@46594:   | Axiom_le of int
huffman@46594:   | Axiom_lt of int
huffman@46594:   | Rational_eq of Rat.rat
huffman@46594:   | Rational_le of Rat.rat
huffman@46594:   | Rational_lt of Rat.rat
huffman@46594:   | Square of FuncUtil.poly
huffman@46594:   | Eqmul of FuncUtil.poly * positivstellensatz
huffman@46594:   | Sum of positivstellensatz * positivstellensatz
huffman@46594:   | Product of positivstellensatz * positivstellensatz;
chaieb@31120: 
huffman@46594:   datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
Philipp@32645: 
huffman@46594:   datatype tree_choice = Left | Right
Philipp@32645: 
huffman@46594:   type prover = tree_choice list ->
huffman@46594:     (thm list * thm list * thm list -> positivstellensatz -> thm) ->
huffman@46594:       thm list * thm list * thm list -> thm * pss_tree
huffman@46594:   type cert_conv = cterm -> thm * pss_tree
Philipp@32645: 
huffman@46594:   val gen_gen_real_arith :
huffman@46594:     Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
huffman@46594:      conv * conv * conv * conv * conv * conv * prover -> cert_conv
huffman@46594:   val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
huffman@46594:     thm list * thm list * thm list -> thm * pss_tree
chaieb@31120: 
huffman@46594:   val gen_real_arith : Proof.context ->
huffman@46594:     (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
Philipp@32645: 
huffman@46594:   val gen_prover_real_arith : Proof.context -> prover -> cert_conv
Philipp@32645: 
huffman@46594:   val is_ratconst : cterm -> bool
huffman@46594:   val dest_ratconst : cterm -> Rat.rat
huffman@46594:   val cterm_of_rat : Rat.rat -> cterm
Philipp@32645: 
chaieb@31120: end
chaieb@31120: 
Philipp@32645: structure RealArith : REAL_ARITH =
chaieb@31120: struct
chaieb@31120: 
huffman@46594: open Conv
chaieb@31120: (* ------------------------------------------------------------------------- *)
chaieb@31120: (* Data structure for Positivstellensatz refutations.                        *)
chaieb@31120: (* ------------------------------------------------------------------------- *)
chaieb@31120: 
chaieb@31120: datatype positivstellensatz =
huffman@46594:     Axiom_eq of int
huffman@46594:   | Axiom_le of int
huffman@46594:   | Axiom_lt of int
huffman@46594:   | Rational_eq of Rat.rat
huffman@46594:   | Rational_le of Rat.rat
huffman@46594:   | Rational_lt of Rat.rat
huffman@46594:   | Square of FuncUtil.poly
huffman@46594:   | Eqmul of FuncUtil.poly * positivstellensatz
huffman@46594:   | Sum of positivstellensatz * positivstellensatz
huffman@46594:   | Product of positivstellensatz * positivstellensatz;
chaieb@31120:          (* Theorems used in the procedure *)
chaieb@31120: 
Philipp@32645: datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
Philipp@32645: datatype tree_choice = Left | Right
huffman@46594: type prover = tree_choice list ->
Philipp@32645:   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
huffman@46594:     thm list * thm list * thm list -> thm * pss_tree
Philipp@32645: type cert_conv = cterm -> thm * pss_tree
chaieb@31120: 
Philipp@32645: 
Philipp@32645:     (* Some useful derived rules *)
huffman@46594: fun deduct_antisym_rule tha thb =
huffman@46594:     Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
wenzelm@36945:      (Thm.implies_intr (cprop_of tha) thb);
Philipp@32645: 
wenzelm@44058: fun prove_hyp tha thb =
wenzelm@44058:   if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
wenzelm@36945:   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
Philipp@32645: 
wenzelm@33443: val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
wenzelm@33443:      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
wenzelm@33443:      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
wenzelm@33443:   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
chaieb@31120: 
chaieb@31120: val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
huffman@46594: val pth_add =
wenzelm@33443:   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
wenzelm@33443:     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
wenzelm@33443:     "(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
wenzelm@33443:     "(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
wenzelm@33443:     "(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
chaieb@31120: 
huffman@46594: val pth_mul =
wenzelm@33443:   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
wenzelm@33443:     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
wenzelm@33443:     "(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
wenzelm@33443:     "(x > 0 ==>  y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
wenzelm@33443:     "(x > 0 ==>  y > 0 ==> x * y > 0)"
chaieb@31120:   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
wenzelm@33443:     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
chaieb@31120: 
chaieb@31120: val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
chaieb@31120: val pth_square = @{lemma "x * x >= (0::real)"  by simp};
chaieb@31120: 
wenzelm@33443: val weak_dnf_simps =
wenzelm@45654:   List.take (@{thms simp_thms}, 34) @
wenzelm@33443:     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
wenzelm@33443:       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
chaieb@31120: 
huffman@44454: (*
wenzelm@33443: val nnfD_simps =
wenzelm@33443:   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
wenzelm@33443:     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
wenzelm@33443:     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
huffman@44454: *)
chaieb@31120: 
chaieb@31120: val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
wenzelm@33443: val prenex_simps =
wenzelm@33443:   map (fn th => th RS sym)
wenzelm@33443:     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
haftmann@37598:       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
chaieb@31120: 
wenzelm@33443: val real_abs_thms1 = @{lemma
wenzelm@33443:   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
wenzelm@33443:   "((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
wenzelm@33443:   "((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
wenzelm@33443:   "((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
wenzelm@33443:   "((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
wenzelm@33443:   "((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
wenzelm@33443:   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
wenzelm@33443:   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
wenzelm@33443:   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
wenzelm@33443:   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
wenzelm@33443:   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
wenzelm@33443:   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
wenzelm@33443:   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
wenzelm@33443:   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
wenzelm@33443:   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
wenzelm@33443:   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
wenzelm@33443:   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
wenzelm@33443:   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
wenzelm@33443:   "((min x y >= r) = (x >= r &  y >= r))" and
wenzelm@33443:   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
wenzelm@33443:   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
wenzelm@33443:   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
wenzelm@33443:   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
wenzelm@33443:   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
wenzelm@33443:   "((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
wenzelm@33443:   "((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
wenzelm@33443:   "((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
wenzelm@33443:   "((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
wenzelm@33443:   "((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
wenzelm@33443:   "((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
wenzelm@33443:   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
wenzelm@33443:   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
wenzelm@33443:   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
wenzelm@33443:   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
wenzelm@33443:   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
wenzelm@33443:   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
wenzelm@33443:   "((min x y > r) = (x > r &  y > r))" and
wenzelm@33443:   "((min x y + a > r) = (a + x > r & a + y > r))" and
wenzelm@33443:   "((a + min x y > r) = (a + x > r & a + y > r))" and
wenzelm@33443:   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
wenzelm@33443:   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
wenzelm@33443:   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
chaieb@31120:   by auto};
chaieb@31120: 
haftmann@35028: val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
chaieb@31120:   by (atomize (full)) (auto split add: abs_split)};
chaieb@31120: 
chaieb@31120: val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
chaieb@31120:   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
chaieb@31120: 
chaieb@31120: val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
chaieb@31120:   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
chaieb@31120: 
chaieb@31120: 
krauss@39920:          (* Miscellaneous *)
huffman@46594: fun literals_conv bops uops cv =
huffman@46594:   let
huffman@46594:     fun h t =
huffman@46594:       case (term_of t) of
huffman@46594:         b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
huffman@46594:       | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
huffman@46594:       | _ => cv t
huffman@46594:   in h end;
chaieb@31120: 
huffman@46594: fun cterm_of_rat x =
huffman@46594:   let
huffman@46594:     val (a, b) = Rat.quotient_of_rat x
huffman@46594:   in
huffman@46594:     if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
huffman@46594:     else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
huffman@46594:       (Numeral.mk_cnumber @{ctyp "real"} a))
huffman@46594:       (Numeral.mk_cnumber @{ctyp "real"} b)
huffman@46594:   end;
chaieb@31120: 
huffman@46594: fun dest_ratconst t =
huffman@46594:   case term_of t of
huffman@46594:     Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
huffman@46594:   | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
huffman@46594: fun is_ratconst t = can dest_ratconst t
chaieb@31120: 
huffman@44454: (*
huffman@46594: fun find_term p t = if p t then t else
chaieb@31120:  case t of
chaieb@31120:   a$b => (find_term p a handle TERM _ => find_term p b)
chaieb@31120:  | Abs (_,_,t') => find_term p t'
chaieb@31120:  | _ => raise TERM ("find_term",[t]);
huffman@44454: *)
chaieb@31120: 
huffman@46594: fun find_cterm p t =
huffman@46594:   if p t then t else
huffman@46594:   case term_of t of
huffman@46594:     _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
huffman@46594:   | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
huffman@46594:   | _ => raise CTERM ("find_cterm",[t]);
chaieb@31120: 
chaieb@31120:     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
chaieb@31120: fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
chaieb@31120: fun is_comb t = case (term_of t) of _$_ => true | _ => false;
chaieb@31120: 
chaieb@31120: fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
chaieb@31120:   handle CTERM _ => false;
chaieb@31120: 
Philipp@32645: 
Philipp@32645: (* Map back polynomials to HOL.                         *)
Philipp@32645: 
huffman@46594: fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
Philipp@32828:   (Numeral.mk_cnumber @{ctyp nat} k)
Philipp@32645: 
huffman@46594: fun cterm_of_monomial m =
huffman@46594:   if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
huffman@46594:   else
huffman@46594:     let
huffman@46594:       val m' = FuncUtil.dest_monomial m
huffman@46594:       val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
huffman@46594:     in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
huffman@46594:     end
Philipp@32645: 
huffman@46594: fun cterm_of_cmonomial (m,c) =
huffman@46594:   if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
huffman@46594:   else if c = Rat.one then cterm_of_monomial m
huffman@46594:   else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
Philipp@32645: 
huffman@46594: fun cterm_of_poly p =
huffman@46594:   if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
huffman@46594:   else
huffman@46594:     let
huffman@46594:       val cms = map cterm_of_cmonomial
huffman@46594:         (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
huffman@46594:     in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
huffman@46594:     end;
Philipp@32645: 
huffman@46594: (* A general real arithmetic prover *)
chaieb@31120: 
chaieb@31120: fun gen_gen_real_arith ctxt (mk_numeric,
chaieb@31120:        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
chaieb@31120:        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
huffman@46594:        absconv1,absconv2,prover) =
huffman@46594:   let
huffman@46594:     val pre_ss = HOL_basic_ss addsimps
huffman@46594:       @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib all_conj_distrib if_bool_eq_disj}
huffman@46594:     val prenex_ss = HOL_basic_ss addsimps prenex_simps
huffman@46594:     val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
huffman@46594:     val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
huffman@46594:     val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
huffman@46594:     val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
huffman@46594:     val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
huffman@46594:     val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
huffman@46594:     fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
huffman@46594:     fun oprconv cv ct =
huffman@46594:       let val g = Thm.dest_fun2 ct
huffman@46594:       in if g aconvc @{cterm "op <= :: real => _"}
huffman@46594:             orelse g aconvc @{cterm "op < :: real => _"}
huffman@46594:          then arg_conv cv ct else arg1_conv cv ct
huffman@46594:       end
chaieb@31120: 
huffman@46594:     fun real_ineq_conv th ct =
huffman@46594:       let
huffman@46594:         val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
huffman@46594:           handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
huffman@46594:       in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
huffman@46594:       end
huffman@46594:     val [real_lt_conv, real_le_conv, real_eq_conv,
huffman@46594:          real_not_lt_conv, real_not_le_conv, _] =
huffman@46594:          map real_ineq_conv pth
huffman@46594:     fun match_mp_rule ths ths' =
huffman@46594:       let
huffman@46594:         fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
huffman@46594:           | th::ths => (ths' MRS th handle THM _ => f ths ths')
huffman@46594:       in f ths ths' end
huffman@46594:     fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
chaieb@31120:          (match_mp_rule pth_mul [th, th'])
huffman@46594:     fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
chaieb@31120:          (match_mp_rule pth_add [th, th'])
huffman@46594:     fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
huffman@46594:        (instantiate' [] [SOME ct] (th RS pth_emul))
huffman@46594:     fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
chaieb@31120:        (instantiate' [] [SOME t] pth_square)
chaieb@31120: 
huffman@46594:     fun hol_of_positivstellensatz(eqs,les,lts) proof =
huffman@46594:       let
huffman@46594:         fun translate prf =
huffman@46594:           case prf of
huffman@46594:             Axiom_eq n => nth eqs n
huffman@46594:           | Axiom_le n => nth les n
huffman@46594:           | Axiom_lt n => nth lts n
huffman@46594:           | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop}
huffman@46594:                           (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x))
chaieb@31120:                                @{cterm "0::real"})))
huffman@46594:           | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop}
huffman@46594:                           (Thm.apply (Thm.apply @{cterm "op <=::real => _"}
chaieb@31120:                                      @{cterm "0::real"}) (mk_numeric x))))
huffman@46594:           | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop}
wenzelm@46497:                       (Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"})
chaieb@31120:                         (mk_numeric x))))
huffman@46594:           | Square pt => square_rule (cterm_of_poly pt)
huffman@46594:           | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
huffman@46594:           | Sum(p1,p2) => add_rule (translate p1) (translate p2)
huffman@46594:           | Product(p1,p2) => mul_rule (translate p1) (translate p2)
huffman@46594:       in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
chaieb@31120:           (translate proof)
huffman@46594:       end
huffman@46594: 
huffman@46594:     val init_conv = presimp_conv then_conv
huffman@46594:         nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
huffman@46594:         weak_dnf_conv
chaieb@31120: 
huffman@46594:     val concl = Thm.dest_arg o cprop_of
huffman@46594:     fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
huffman@46594:     val is_req = is_binop @{cterm "op =:: real => _"}
huffman@46594:     val is_ge = is_binop @{cterm "op <=:: real => _"}
huffman@46594:     val is_gt = is_binop @{cterm "op <:: real => _"}
huffman@46594:     val is_conj = is_binop @{cterm HOL.conj}
huffman@46594:     val is_disj = is_binop @{cterm HOL.disj}
huffman@46594:     fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
huffman@46594:     fun disj_cases th th1 th2 =
huffman@46594:       let
huffman@46594:         val (p,q) = Thm.dest_binop (concl th)
huffman@46594:         val c = concl th1
huffman@46594:         val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
huffman@46594:       in Thm.implies_elim (Thm.implies_elim
wenzelm@36945:           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
wenzelm@46497:           (Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
wenzelm@46497:         (Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2)
huffman@46594:       end
huffman@46594:     fun overall cert_choice dun ths =
huffman@46594:       case ths of
huffman@46594:         [] =>
huffman@46594:         let
huffman@46594:           val (eq,ne) = List.partition (is_req o concl) dun
huffman@46594:           val (le,nl) = List.partition (is_ge o concl) ne
huffman@46594:           val lt = filter (is_gt o concl) nl
huffman@46594:         in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
huffman@46594:       | th::oths =>
huffman@46594:         let
huffman@46594:           val ct = concl th
huffman@46594:         in
huffman@46594:           if is_conj ct then
huffman@46594:             let
huffman@46594:               val (th1,th2) = conj_pair th
huffman@46594:             in overall cert_choice dun (th1::th2::oths) end
huffman@46594:           else if is_disj ct then
huffman@46594:             let
huffman@46594:               val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
huffman@46594:               val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
huffman@46594:             in (disj_cases th th1 th2, Branch (cert1, cert2)) end
huffman@46594:           else overall cert_choice (th::dun) oths
huffman@46594:         end
huffman@46594:     fun dest_binary b ct =
huffman@46594:         if is_binop b ct then Thm.dest_binop ct
huffman@46594:         else raise CTERM ("dest_binary",[b,ct])
huffman@46594:     val dest_eq = dest_binary @{cterm "op = :: real => _"}
huffman@46594:     val neq_th = nth pth 5
huffman@46594:     fun real_not_eq_conv ct =
huffman@46594:       let
huffman@46594:         val (l,r) = dest_eq (Thm.dest_arg ct)
huffman@46594:         val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
huffman@46594:         val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
huffman@46594:         val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
huffman@46594:         val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
huffman@46594:         val th' = Drule.binop_cong_rule @{cterm HOL.disj}
huffman@46594:           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
huffman@46594:           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
huffman@46594:       in Thm.transitive th th'
huffman@46594:       end
huffman@46594:     fun equal_implies_1_rule PQ =
huffman@46594:       let
huffman@46594:         val P = Thm.lhs_of PQ
huffman@46594:       in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
huffman@46594:       end
huffman@46594:     (* FIXME!!! Copied from groebner.ml *)
huffman@46594:     val strip_exists =
huffman@46594:       let
huffman@46594:         fun h (acc, t) =
huffman@46594:           case (term_of t) of
huffman@46594:             Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
huffman@46594:           | _ => (acc,t)
huffman@46594:       in fn t => h ([],t)
huffman@46594:       end
huffman@46594:     fun name_of x =
huffman@46594:       case term_of x of
huffman@46594:         Free(s,_) => s
huffman@46594:       | Var ((s,_),_) => s
huffman@46594:       | _ => "x"
chaieb@31120: 
huffman@46594:     fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
chaieb@31120: 
huffman@46594:     val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
chaieb@31120: 
huffman@46594:     fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
huffman@46594:     fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
chaieb@31120: 
huffman@46594:     fun choose v th th' =
huffman@46594:       case concl_of th of
huffman@46594:         @{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
huffman@46594:         let
huffman@46594:           val p = (funpow 2 Thm.dest_arg o cprop_of) th
huffman@46594:           val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
huffman@46594:           val th0 = fconv_rule (Thm.beta_conversion true)
huffman@46594:             (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
huffman@46594:           val pv = (Thm.rhs_of o Thm.beta_conversion true)
huffman@46594:             (Thm.apply @{cterm Trueprop} (Thm.apply p v))
huffman@46594:           val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
huffman@46594:         in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
huffman@46594:       | _ => raise THM ("choose",0,[th, th'])
chaieb@31120: 
huffman@46594:     fun simple_choose v th =
huffman@46594:       choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
chaieb@31120: 
huffman@46594:     val strip_forall =
huffman@46594:       let
huffman@46594:         fun h (acc, t) =
huffman@46594:           case (term_of t) of
huffman@46594:             Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
huffman@46594:           | _ => (acc,t)
huffman@46594:       in fn t => h ([],t)
huffman@46594:       end
chaieb@31120: 
huffman@46594:     fun f ct =
huffman@46594:       let
huffman@46594:         val nnf_norm_conv' =
huffman@46594:           nnf_conv then_conv
huffman@46594:           literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
huffman@46594:           (Conv.cache_conv
huffman@46594:             (first_conv [real_lt_conv, real_le_conv,
huffman@46594:                          real_eq_conv, real_not_lt_conv,
huffman@46594:                          real_not_le_conv, real_not_eq_conv, all_conv]))
huffman@46594:         fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
huffman@46594:                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
huffman@46594:                   try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
huffman@46594:         val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
huffman@46594:         val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
huffman@46594:         val tm0 = Thm.dest_arg (Thm.rhs_of th0)
huffman@46594:         val (th, certificates) =
huffman@46594:           if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
huffman@46594:           let
huffman@46594:             val (evs,bod) = strip_exists tm0
huffman@46594:             val (avs,ibod) = strip_forall bod
huffman@46594:             val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
huffman@46594:             val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
huffman@46594:             val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
huffman@46594:           in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
huffman@46594:           end
huffman@46594:       in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
huffman@46594:       end
huffman@46594:   in f
huffman@46594:   end;
chaieb@31120: 
chaieb@31120: (* A linear arithmetic prover *)
chaieb@31120: local
Philipp@32828:   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
haftmann@39027:   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
chaieb@31120:   val one_tm = @{cterm "1::real"}
Philipp@32829:   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
haftmann@33038:      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
Philipp@32829:        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
chaieb@31120: 
huffman@46594:   fun linear_ineqs vars (les,lts) =
huffman@46594:     case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
huffman@46594:       SOME r => r
huffman@46594:     | NONE =>
huffman@46594:       (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
huffman@46594:          SOME r => r
huffman@46594:        | NONE =>
huffman@46594:          if null vars then error "linear_ineqs: no contradiction" else
huffman@46594:          let
huffman@46594:            val ineqs = les @ lts
huffman@46594:            fun blowup v =
huffman@46594:              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
huffman@46594:              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
huffman@46594:              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
huffman@46594:            val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
huffman@46594:              (map (fn v => (v,blowup v)) vars)))
huffman@46594:            fun addup (e1,p1) (e2,p2) acc =
huffman@46594:              let
huffman@46594:                val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
huffman@46594:                val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
huffman@46594:              in
huffman@46594:                if c1 */ c2 >=/ Rat.zero then acc else
huffman@46594:                let
huffman@46594:                  val e1' = linear_cmul (Rat.abs c2) e1
huffman@46594:                  val e2' = linear_cmul (Rat.abs c1) e2
huffman@46594:                  val p1' = Product(Rational_lt(Rat.abs c2),p1)
huffman@46594:                  val p2' = Product(Rational_lt(Rat.abs c1),p2)
huffman@46594:                in (linear_add e1' e2',Sum(p1',p2'))::acc
huffman@46594:                end
huffman@46594:              end
huffman@46594:            val (les0,les1) =
huffman@46594:              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
huffman@46594:            val (lts0,lts1) =
huffman@46594:              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
huffman@46594:            val (lesp,lesn) =
huffman@46594:              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
huffman@46594:            val (ltsp,ltsn) =
huffman@46594:              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
huffman@46594:            val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
huffman@46594:            val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
chaieb@31120:                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
huffman@46594:          in linear_ineqs (remove (op aconvc) v vars) (les',lts')
huffman@46594:          end)
chaieb@31120: 
huffman@46594:   fun linear_eqs(eqs,les,lts) =
huffman@46594:     case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
huffman@46594:       SOME r => r
huffman@46594:     | NONE =>
huffman@46594:       (case eqs of
huffman@46594:          [] =>
huffman@46594:          let val vars = remove (op aconvc) one_tm
huffman@46594:              (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
huffman@46594:          in linear_ineqs vars (les,lts) end
huffman@46594:        | (e,p)::es =>
huffman@46594:          if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
huffman@46594:          let
huffman@46594:            val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
huffman@46594:            fun xform (inp as (t,q)) =
huffman@46594:              let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
huffman@46594:                if d =/ Rat.zero then inp else
huffman@46594:                let
huffman@46594:                  val k = (Rat.neg d) */ Rat.abs c // c
huffman@46594:                  val e' = linear_cmul k e
huffman@46594:                  val t' = linear_cmul (Rat.abs c) t
huffman@46594:                  val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
huffman@46594:                  val q' = Product(Rational_lt(Rat.abs c),q)
huffman@46594:                in (linear_add e' t',Sum(p',q'))
huffman@46594:                end
huffman@46594:              end
huffman@46594:          in linear_eqs(map xform es,map xform les,map xform lts)
huffman@46594:          end)
chaieb@31120: 
huffman@46594:   fun linear_prover (eq,le,lt) =
huffman@46594:     let
huffman@46594:       val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
huffman@46594:       val les = map_index (fn (n, p) => (p,Axiom_le n)) le
huffman@46594:       val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
huffman@46594:     in linear_eqs(eqs,les,lts)
chaieb@31120:     end
chaieb@31120: 
huffman@46594:   fun lin_of_hol ct =
huffman@46594:     if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
huffman@46594:     else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
huffman@46594:     else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
huffman@46594:     else
huffman@46594:       let val (lop,r) = Thm.dest_comb ct
huffman@46594:       in
huffman@46594:         if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
huffman@46594:         else
huffman@46594:           let val (opr,l) = Thm.dest_comb lop
huffman@46594:           in
huffman@46594:             if opr aconvc @{cterm "op + :: real =>_"}
huffman@46594:             then linear_add (lin_of_hol l) (lin_of_hol r)
huffman@46594:             else if opr aconvc @{cterm "op * :: real =>_"}
huffman@46594:                     andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
huffman@46594:             else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
huffman@46594:           end
huffman@46594:       end
huffman@46594: 
huffman@46594:   fun is_alien ct =
huffman@46594:       case term_of ct of
huffman@46594:         Const(@{const_name "real"}, _)$ n =>
huffman@46594:         if can HOLogic.dest_number n then false else true
huffman@46594:       | _ => false
huffman@46594: in
huffman@46594: fun real_linear_prover translator (eq,le,lt) =
huffman@46594:   let
huffman@46594:     val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
huffman@46594:     val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
huffman@46594:     val eq_pols = map lhs eq
huffman@46594:     val le_pols = map rhs le
huffman@46594:     val lt_pols = map rhs lt
huffman@46594:     val aliens = filter is_alien
huffman@46594:       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
huffman@46594:                 (eq_pols @ le_pols @ lt_pols) [])
huffman@46594:     val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
huffman@46594:     val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
huffman@46594:     val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
huffman@46594:   in ((translator (eq,le',lt) proof), Trivial)
huffman@46594:   end
chaieb@31120: end;
chaieb@31120: 
chaieb@31120: (* A less general generic arithmetic prover dealing with abs,max and min*)
chaieb@31120: 
chaieb@31120: local
huffman@46594:   val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
huffman@46594:   fun absmaxmin_elim_conv1 ctxt =
chaieb@31120:     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
chaieb@31120: 
huffman@46594:   val absmaxmin_elim_conv2 =
huffman@46594:     let
huffman@46594:       val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
huffman@46594:       val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
huffman@46594:       val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
huffman@46594:       val abs_tm = @{cterm "abs :: real => _"}
huffman@46594:       val p_tm = @{cpat "?P :: real => bool"}
huffman@46594:       val x_tm = @{cpat "?x :: real"}
huffman@46594:       val y_tm = @{cpat "?y::real"}
huffman@46594:       val is_max = is_binop @{cterm "max :: real => _"}
huffman@46594:       val is_min = is_binop @{cterm "min :: real => _"}
huffman@46594:       fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
huffman@46594:       fun eliminate_construct p c tm =
huffman@46594:         let
huffman@46594:           val t = find_cterm p tm
huffman@46594:           val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
huffman@46594:           val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
huffman@46594:         in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
huffman@46594:                      (Thm.transitive th0 (c p ax))
huffman@46594:         end
chaieb@31120: 
huffman@46594:       val elim_abs = eliminate_construct is_abs
huffman@46594:         (fn p => fn ax =>
huffman@46594:           Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
huffman@46594:       val elim_max = eliminate_construct is_max
huffman@46594:         (fn p => fn ax =>
huffman@46594:           let val (ax,y) = Thm.dest_comb ax
huffman@46594:           in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
huffman@46594:                              pth_max end)
huffman@46594:       val elim_min = eliminate_construct is_min
huffman@46594:         (fn p => fn ax =>
huffman@46594:           let val (ax,y) = Thm.dest_comb ax
huffman@46594:           in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
huffman@46594:                              pth_min end)
huffman@46594:     in first_conv [elim_abs, elim_max, elim_min, all_conv]
huffman@46594:     end;
huffman@46594: in
huffman@46594: fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
huffman@46594:   gen_gen_real_arith ctxt
huffman@46594:     (mkconst,eq,ge,gt,norm,neg,add,mul,
huffman@46594:      absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
chaieb@31120: end;
chaieb@31120: 
huffman@46594: (* An instance for reals*)
chaieb@31120: 
huffman@46594: fun gen_prover_real_arith ctxt prover =
huffman@46594:   let
huffman@46594:     fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
huffman@46594:     val {add, mul, neg, pow = _, sub = _, main} =
huffman@46594:         Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
huffman@46594:         (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
huffman@46594:         simple_cterm_ord
huffman@46594:   in gen_real_arith ctxt
huffman@46594:      (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
huffman@46594:       main,neg,add,mul, prover)
huffman@46594:   end;
chaieb@31120: 
chaieb@31120: end