src/HOL/Library/positivstellensatz.ML
author haftmann
Wed Oct 21 12:02:56 2009 +0200 (2009-10-21)
changeset 33042 ddf1f03a9ad9
parent 33039 5018f6a76b3f
child 33063 4d462963a7db
permissions -rw-r--r--
curried union as canonical list operation
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(* Title:      Library/Sum_Of_Squares/positivstellensatz
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   Author:     Amine Chaieb, University of Cambridge
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   Description: A generic arithmetic prover based on Positivstellensatz certificates --- 
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    also implements Fourrier-Motzkin elimination as a special case Fourrier-Motzkin elimination.
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*)
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(* A functor for finite mappings based on Tables *)
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signature FUNC = 
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sig
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 include TABLE
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 val apply : 'a table -> key -> 'a
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 val applyd :'a table -> (key -> 'a) -> key -> 'a
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 val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
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 val dom : 'a table -> key list
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 val tryapplyd : 'a table -> key -> 'a -> 'a
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 val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
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 val choose : 'a table -> key * 'a
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 val onefunc : key * 'a -> 'a table
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end;
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functor FuncFun(Key: KEY) : FUNC=
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struct
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structure Tab = Table(Key);
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open Tab;
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fun dom a = sort Key.ord (Tab.keys a);
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fun applyd f d x = case Tab.lookup f x of 
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   SOME y => y
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 | NONE => d x;
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fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
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fun tryapplyd f a d = applyd f (K d) a;
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fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
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fun combine f z a b = 
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 let
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  fun h (k,v) t = case Tab.lookup t k of
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     NONE => Tab.update (k,v) t
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   | SOME v' => let val w = f v v'
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     in if z w then Tab.delete k t else Tab.update (k,w) t end;
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  in Tab.fold h a b end;
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fun choose f = case Tab.min_key f of 
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   SOME k => (k, the (Tab.lookup f k))
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 | NONE => error "FuncFun.choose : Completely empty function"
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fun onefunc kv = update kv empty
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end;
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(* Some standard functors and utility functions for them *)
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structure FuncUtil =
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struct
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structure Intfunc = FuncFun(type key = int val ord = int_ord);
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structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
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structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
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structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
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structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
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val cterm_ord = TermOrd.fast_term_ord o pairself term_of
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structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
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type monomial = int Ctermfunc.table;
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val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
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structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
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type poly = Rat.rat Monomialfunc.table;
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(* The ordering so we can create canonical HOL polynomials.                  *)
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fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
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fun monomial_order (m1,m2) =
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 if Ctermfunc.is_empty m2 then LESS 
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 else if Ctermfunc.is_empty m1 then GREATER 
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 else
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  let val mon1 = dest_monomial m1 
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      val mon2 = dest_monomial m2
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      val deg1 = fold (Integer.add o snd) mon1 0
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      val deg2 = fold (Integer.add o snd) mon2 0 
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  in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
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     else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
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  end;
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end
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(* positivstellensatz datatype and prover generation *)
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signature REAL_ARITH = 
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sig
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  datatype positivstellensatz =
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   Axiom_eq of int
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 | Axiom_le of int
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 | Axiom_lt of int
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 | Rational_eq of Rat.rat
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 | Rational_le of Rat.rat
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 | Rational_lt of Rat.rat
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 | Square of FuncUtil.poly
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 | Eqmul of FuncUtil.poly * positivstellensatz
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 | Sum of positivstellensatz * positivstellensatz
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 | Product of positivstellensatz * positivstellensatz;
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datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
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datatype tree_choice = Left | Right
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type prover = tree_choice list -> 
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  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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  thm list * thm list * thm list -> thm * pss_tree
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type cert_conv = cterm -> thm * pss_tree
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val gen_gen_real_arith :
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  Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
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   conv * conv * conv * conv * conv * conv * prover -> cert_conv
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val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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  thm list * thm list * thm list -> thm * pss_tree
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val gen_real_arith : Proof.context ->
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  (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
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val gen_prover_real_arith : Proof.context -> prover -> cert_conv
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val is_ratconst : cterm -> bool
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val dest_ratconst : cterm -> Rat.rat
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val cterm_of_rat : Rat.rat -> cterm
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end
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structure RealArith : REAL_ARITH =
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struct
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 open Conv
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(* ------------------------------------------------------------------------- *)
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(* Data structure for Positivstellensatz refutations.                        *)
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(* ------------------------------------------------------------------------- *)
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datatype positivstellensatz =
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   Axiom_eq of int
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 | Axiom_le of int
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 | Axiom_lt of int
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 | Rational_eq of Rat.rat
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 | Rational_le of Rat.rat
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 | Rational_lt of Rat.rat
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 | Square of FuncUtil.poly
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 | Eqmul of FuncUtil.poly * positivstellensatz
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 | Sum of positivstellensatz * positivstellensatz
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 | Product of positivstellensatz * positivstellensatz;
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         (* Theorems used in the procedure *)
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datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
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datatype tree_choice = Left | Right
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type prover = tree_choice list -> 
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  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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  thm list * thm list * thm list -> thm * pss_tree
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type cert_conv = cterm -> thm * pss_tree
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val my_eqs = Unsynchronized.ref ([] : thm list);
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val my_les = Unsynchronized.ref ([] : thm list);
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val my_lts = Unsynchronized.ref ([] : thm list);
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val my_proof = Unsynchronized.ref (Axiom_eq 0);
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val my_context = Unsynchronized.ref @{context};
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val my_mk_numeric = Unsynchronized.ref ((K @{cterm True}) :Rat.rat -> cterm);
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val my_numeric_eq_conv = Unsynchronized.ref no_conv;
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val my_numeric_ge_conv = Unsynchronized.ref no_conv;
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val my_numeric_gt_conv = Unsynchronized.ref no_conv;
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val my_poly_conv = Unsynchronized.ref no_conv;
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val my_poly_neg_conv = Unsynchronized.ref no_conv;
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val my_poly_add_conv = Unsynchronized.ref no_conv;
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val my_poly_mul_conv = Unsynchronized.ref no_conv;
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    (* Some useful derived rules *)
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fun deduct_antisym_rule tha thb = 
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    equal_intr (implies_intr (cprop_of thb) tha) 
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     (implies_intr (cprop_of tha) thb);
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fun prove_hyp tha thb = 
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  if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) 
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  then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
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fun conjunctions th = case try Conjunction.elim th of
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   SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
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 | NONE => [th];
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val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) 
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     &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
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     &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
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  by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> 
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conjunctions;
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val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
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val pth_add = 
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 @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) 
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    &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) 
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    &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) 
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    &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) 
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    &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
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val pth_mul = 
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  @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& 
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           (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& 
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           (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
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           (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
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           (x > 0 ==>  y > 0 ==> x * y > 0)"
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  by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
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    mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
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val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
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val pth_square = @{lemma "x * x >= (0::real)"  by simp};
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val weak_dnf_simps = List.take (simp_thms, 34) 
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    @ conjunctions @{lemma "((P & (Q | R)) = ((P&Q) | (P&R))) &&& ((Q | R) & P) = ((Q&P) | (R&P)) &&& (P & Q) = (Q & P) &&& ((P | Q) = (Q | P))" by blast+};
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val nnfD_simps = conjunctions @{lemma "((~(P & Q)) = (~P | ~Q)) &&& ((~(P | Q)) = (~P & ~Q) ) &&& ((P --> Q) = (~P | Q) ) &&& ((P = Q) = ((P & Q) | (~P & ~ Q))) &&& ((~(P = Q)) = ((P & ~ Q) | (~P & Q)) ) &&& ((~ ~(P)) = P)" by blast+}
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val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
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val prenex_simps = map (fn th => th RS sym) ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ @{thms "all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
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val real_abs_thms1 = conjunctions @{lemma
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  "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
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  ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
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  ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
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  ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
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  ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
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  ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
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  ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
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  ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
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  ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
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  ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
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  ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
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  ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
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  ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
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  ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
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  ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
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  ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
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  ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
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  ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
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  ((min x y >= r) = (x >= r &  y >= r)) &&&
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  ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
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  ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
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  ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
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  ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
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  ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
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  ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
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  ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
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  ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
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  ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
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  ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
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  ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
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  ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
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  ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
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  ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
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  ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
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  ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
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  ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
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  ((min x y > r) = (x > r &  y > r)) &&&
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  ((min x y + a > r) = (a + x > r & a + y > r)) &&&
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  ((a + min x y > r) = (a + x > r & a + y > r)) &&&
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  ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
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  ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
chaieb@31120
   270
  ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
chaieb@31120
   271
  by auto};
chaieb@31120
   272
chaieb@31120
   273
val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
chaieb@31120
   274
  by (atomize (full)) (auto split add: abs_split)};
chaieb@31120
   275
chaieb@31120
   276
val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
chaieb@31120
   277
  by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
chaieb@31120
   278
chaieb@31120
   279
val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
chaieb@31120
   280
  by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
chaieb@31120
   281
chaieb@31120
   282
chaieb@31120
   283
         (* Miscalineous *)
chaieb@31120
   284
fun literals_conv bops uops cv = 
chaieb@31120
   285
 let fun h t =
chaieb@31120
   286
  case (term_of t) of 
chaieb@31120
   287
   b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
chaieb@31120
   288
 | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
chaieb@31120
   289
 | _ => cv t
chaieb@31120
   290
 in h end;
chaieb@31120
   291
chaieb@31120
   292
fun cterm_of_rat x = 
chaieb@31120
   293
let val (a, b) = Rat.quotient_of_rat x
chaieb@31120
   294
in 
chaieb@31120
   295
 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
chaieb@31120
   296
  else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
chaieb@31120
   297
                   (Numeral.mk_cnumber @{ctyp "real"} a))
chaieb@31120
   298
        (Numeral.mk_cnumber @{ctyp "real"} b)
chaieb@31120
   299
end;
chaieb@31120
   300
chaieb@31120
   301
  fun dest_ratconst t = case term_of t of
chaieb@31120
   302
   Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
chaieb@31120
   303
 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
chaieb@31120
   304
 fun is_ratconst t = can dest_ratconst t
chaieb@31120
   305
chaieb@31120
   306
fun find_term p t = if p t then t else 
chaieb@31120
   307
 case t of
chaieb@31120
   308
  a$b => (find_term p a handle TERM _ => find_term p b)
chaieb@31120
   309
 | Abs (_,_,t') => find_term p t'
chaieb@31120
   310
 | _ => raise TERM ("find_term",[t]);
chaieb@31120
   311
chaieb@31120
   312
fun find_cterm p t = if p t then t else 
chaieb@31120
   313
 case term_of t of
chaieb@31120
   314
  a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
chaieb@31120
   315
 | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
chaieb@31120
   316
 | _ => raise CTERM ("find_cterm",[t]);
chaieb@31120
   317
chaieb@31120
   318
    (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
chaieb@31120
   319
fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
chaieb@31120
   320
fun is_comb t = case (term_of t) of _$_ => true | _ => false;
chaieb@31120
   321
chaieb@31120
   322
fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
chaieb@31120
   323
  handle CTERM _ => false;
chaieb@31120
   324
Philipp@32645
   325
Philipp@32645
   326
(* Map back polynomials to HOL.                         *)
Philipp@32645
   327
Philipp@32828
   328
fun cterm_of_varpow x k = if k = 1 then x else Thm.capply (Thm.capply @{cterm "op ^ :: real => _"} x) 
Philipp@32828
   329
  (Numeral.mk_cnumber @{ctyp nat} k)
Philipp@32645
   330
Philipp@32645
   331
fun cterm_of_monomial m = 
Philipp@32829
   332
 if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"} 
Philipp@32645
   333
 else 
Philipp@32645
   334
  let 
Philipp@32828
   335
   val m' = FuncUtil.dest_monomial m
Philipp@32645
   336
   val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 
Philipp@32828
   337
  in foldr1 (fn (s, t) => Thm.capply (Thm.capply @{cterm "op * :: real => _"} s) t) vps
Philipp@32645
   338
  end
Philipp@32645
   339
Philipp@32829
   340
fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
Philipp@32645
   341
    else if c = Rat.one then cterm_of_monomial m
Philipp@32828
   342
    else Thm.capply (Thm.capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
Philipp@32645
   343
Philipp@32645
   344
fun cterm_of_poly p = 
Philipp@32829
   345
 if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"} 
Philipp@32645
   346
 else
Philipp@32645
   347
  let 
Philipp@32645
   348
   val cms = map cterm_of_cmonomial
Philipp@32829
   349
     (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
Philipp@32828
   350
  in foldr1 (fn (t1, t2) => Thm.capply(Thm.capply @{cterm "op + :: real => _"} t1) t2) cms
Philipp@32645
   351
  end;
Philipp@32645
   352
chaieb@31120
   353
    (* A general real arithmetic prover *)
chaieb@31120
   354
chaieb@31120
   355
fun gen_gen_real_arith ctxt (mk_numeric,
chaieb@31120
   356
       numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
chaieb@31120
   357
       poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
chaieb@31120
   358
       absconv1,absconv2,prover) = 
chaieb@31120
   359
let
chaieb@31120
   360
 val _ = my_context := ctxt 
chaieb@31120
   361
 val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ; 
chaieb@31120
   362
          my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
chaieb@31120
   363
          my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv; 
chaieb@31120
   364
          my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
chaieb@31120
   365
 val pre_ss = HOL_basic_ss addsimps simp_thms@ ex_simps@ all_simps@[@{thm not_all},@{thm not_ex},ex_disj_distrib, all_conj_distrib, @{thm if_bool_eq_disj}]
chaieb@31120
   366
 val prenex_ss = HOL_basic_ss addsimps prenex_simps
chaieb@31120
   367
 val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
chaieb@31120
   368
 val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
chaieb@31120
   369
 val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
chaieb@31120
   370
 val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
chaieb@31120
   371
 val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
chaieb@31120
   372
 val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
chaieb@31120
   373
 fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
chaieb@31120
   374
 fun oprconv cv ct = 
chaieb@31120
   375
  let val g = Thm.dest_fun2 ct
chaieb@31120
   376
  in if g aconvc @{cterm "op <= :: real => _"} 
chaieb@31120
   377
       orelse g aconvc @{cterm "op < :: real => _"} 
chaieb@31120
   378
     then arg_conv cv ct else arg1_conv cv ct
chaieb@31120
   379
  end
chaieb@31120
   380
chaieb@31120
   381
 fun real_ineq_conv th ct =
chaieb@31120
   382
  let
Philipp@32828
   383
   val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th 
chaieb@31120
   384
      handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
chaieb@31120
   385
  in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
chaieb@31120
   386
  end 
chaieb@31120
   387
  val [real_lt_conv, real_le_conv, real_eq_conv,
chaieb@31120
   388
       real_not_lt_conv, real_not_le_conv, _] =
chaieb@31120
   389
       map real_ineq_conv pth
chaieb@31120
   390
  fun match_mp_rule ths ths' = 
chaieb@31120
   391
   let
chaieb@31120
   392
     fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
chaieb@31120
   393
      | th::ths => (ths' MRS th handle THM _ => f ths ths')
chaieb@31120
   394
   in f ths ths' end
chaieb@31120
   395
  fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
chaieb@31120
   396
         (match_mp_rule pth_mul [th, th'])
chaieb@31120
   397
  fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
chaieb@31120
   398
         (match_mp_rule pth_add [th, th'])
chaieb@31120
   399
  fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
chaieb@31120
   400
       (instantiate' [] [SOME ct] (th RS pth_emul)) 
chaieb@31120
   401
  fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
chaieb@31120
   402
       (instantiate' [] [SOME t] pth_square)
chaieb@31120
   403
chaieb@31120
   404
  fun hol_of_positivstellensatz(eqs,les,lts) proof =
chaieb@31120
   405
   let 
chaieb@31120
   406
    val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
chaieb@31120
   407
    fun translate prf = case prf of
chaieb@31120
   408
        Axiom_eq n => nth eqs n
chaieb@31120
   409
      | Axiom_le n => nth les n
chaieb@31120
   410
      | Axiom_lt n => nth lts n
Philipp@32828
   411
      | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.capply @{cterm Trueprop} 
Philipp@32828
   412
                          (Thm.capply (Thm.capply @{cterm "op =::real => _"} (mk_numeric x)) 
chaieb@31120
   413
                               @{cterm "0::real"})))
Philipp@32828
   414
      | Rational_le x => eqT_elim(numeric_ge_conv(Thm.capply @{cterm Trueprop} 
Philipp@32828
   415
                          (Thm.capply (Thm.capply @{cterm "op <=::real => _"} 
chaieb@31120
   416
                                     @{cterm "0::real"}) (mk_numeric x))))
Philipp@32828
   417
      | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.capply @{cterm Trueprop} 
Philipp@32828
   418
                      (Thm.capply (Thm.capply @{cterm "op <::real => _"} @{cterm "0::real"})
chaieb@31120
   419
                        (mk_numeric x))))
Philipp@32645
   420
      | Square pt => square_rule (cterm_of_poly pt)
Philipp@32645
   421
      | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
chaieb@31120
   422
      | Sum(p1,p2) => add_rule (translate p1) (translate p2)
chaieb@31120
   423
      | Product(p1,p2) => mul_rule (translate p1) (translate p2)
chaieb@31120
   424
   in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
chaieb@31120
   425
          (translate proof)
chaieb@31120
   426
   end
chaieb@31120
   427
  
chaieb@31120
   428
  val init_conv = presimp_conv then_conv
chaieb@31120
   429
      nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
chaieb@31120
   430
      weak_dnf_conv
chaieb@31120
   431
Philipp@32828
   432
  val concl = Thm.dest_arg o cprop_of
Philipp@32828
   433
  fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
chaieb@31120
   434
  val is_req = is_binop @{cterm "op =:: real => _"}
chaieb@31120
   435
  val is_ge = is_binop @{cterm "op <=:: real => _"}
chaieb@31120
   436
  val is_gt = is_binop @{cterm "op <:: real => _"}
chaieb@31120
   437
  val is_conj = is_binop @{cterm "op &"}
chaieb@31120
   438
  val is_disj = is_binop @{cterm "op |"}
chaieb@31120
   439
  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
chaieb@31120
   440
  fun disj_cases th th1 th2 = 
Philipp@32828
   441
   let val (p,q) = Thm.dest_binop (concl th)
chaieb@31120
   442
       val c = concl th1
chaieb@31120
   443
       val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
Philipp@32828
   444
   in implies_elim (implies_elim
Philipp@32828
   445
          (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
Philipp@32828
   446
          (implies_intr (Thm.capply @{cterm Trueprop} p) th1))
Philipp@32828
   447
        (implies_intr (Thm.capply @{cterm Trueprop} q) th2)
chaieb@31120
   448
   end
Philipp@32645
   449
 fun overall cert_choice dun ths = case ths of
chaieb@31120
   450
  [] =>
chaieb@31120
   451
   let 
chaieb@31120
   452
    val (eq,ne) = List.partition (is_req o concl) dun
chaieb@31120
   453
     val (le,nl) = List.partition (is_ge o concl) ne
chaieb@31120
   454
     val lt = filter (is_gt o concl) nl 
Philipp@32645
   455
    in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
chaieb@31120
   456
 | th::oths =>
chaieb@31120
   457
   let 
chaieb@31120
   458
    val ct = concl th 
chaieb@31120
   459
   in 
chaieb@31120
   460
    if is_conj ct  then
chaieb@31120
   461
     let 
chaieb@31120
   462
      val (th1,th2) = conj_pair th in
Philipp@32645
   463
      overall cert_choice dun (th1::th2::oths) end
chaieb@31120
   464
    else if is_disj ct then
chaieb@31120
   465
      let 
Philipp@32828
   466
       val (th1, cert1) = overall (Left::cert_choice) dun (assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
Philipp@32828
   467
       val (th2, cert2) = overall (Right::cert_choice) dun (assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
Philipp@32645
   468
      in (disj_cases th th1 th2, Branch (cert1, cert2)) end
Philipp@32645
   469
   else overall cert_choice (th::dun) oths
chaieb@31120
   470
  end
Philipp@32828
   471
  fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct 
chaieb@31120
   472
                         else raise CTERM ("dest_binary",[b,ct])
chaieb@31120
   473
  val dest_eq = dest_binary @{cterm "op = :: real => _"}
chaieb@31120
   474
  val neq_th = nth pth 5
chaieb@31120
   475
  fun real_not_eq_conv ct = 
chaieb@31120
   476
   let 
Philipp@32828
   477
    val (l,r) = dest_eq (Thm.dest_arg ct)
Philipp@32828
   478
    val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
Philipp@32828
   479
    val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
chaieb@31120
   480
    val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
chaieb@31120
   481
    val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
chaieb@31120
   482
    val th' = Drule.binop_cong_rule @{cterm "op |"} 
Philipp@32828
   483
     (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
Philipp@32828
   484
     (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
chaieb@31120
   485
    in transitive th th' 
chaieb@31120
   486
  end
chaieb@31120
   487
 fun equal_implies_1_rule PQ = 
chaieb@31120
   488
  let 
Philipp@32828
   489
   val P = Thm.lhs_of PQ
chaieb@31120
   490
  in implies_intr P (equal_elim PQ (assume P))
chaieb@31120
   491
  end
chaieb@31120
   492
 (* FIXME!!! Copied from groebner.ml *)
chaieb@31120
   493
 val strip_exists =
chaieb@31120
   494
  let fun h (acc, t) =
chaieb@31120
   495
   case (term_of t) of
Philipp@32828
   496
    Const("Ex",_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
chaieb@31120
   497
  | _ => (acc,t)
chaieb@31120
   498
  in fn t => h ([],t)
chaieb@31120
   499
  end
chaieb@31120
   500
  fun name_of x = case term_of x of
chaieb@31120
   501
   Free(s,_) => s
chaieb@31120
   502
 | Var ((s,_),_) => s
chaieb@31120
   503
 | _ => "x"
chaieb@31120
   504
chaieb@31120
   505
  fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (abstract_rule (name_of x) x th)
chaieb@31120
   506
chaieb@31120
   507
  val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
chaieb@31120
   508
chaieb@31120
   509
 fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
chaieb@31120
   510
 fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
chaieb@31120
   511
chaieb@31120
   512
 fun choose v th th' = case concl_of th of 
chaieb@31120
   513
   @{term Trueprop} $ (Const("Ex",_)$_) => 
chaieb@31120
   514
    let
chaieb@31120
   515
     val p = (funpow 2 Thm.dest_arg o cprop_of) th
chaieb@31120
   516
     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
chaieb@31120
   517
     val th0 = fconv_rule (Thm.beta_conversion true)
chaieb@31120
   518
         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
chaieb@31120
   519
     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
chaieb@31120
   520
           (Thm.capply @{cterm Trueprop} (Thm.capply p v))
chaieb@31120
   521
     val th1 = forall_intr v (implies_intr pv th')
chaieb@31120
   522
    in implies_elim (implies_elim th0 th) th1  end
chaieb@31120
   523
 | _ => raise THM ("choose",0,[th, th'])
chaieb@31120
   524
chaieb@31120
   525
  fun simple_choose v th = 
chaieb@31120
   526
     choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
chaieb@31120
   527
chaieb@31120
   528
 val strip_forall =
chaieb@31120
   529
  let fun h (acc, t) =
chaieb@31120
   530
   case (term_of t) of
Philipp@32828
   531
    Const("All",_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
chaieb@31120
   532
  | _ => (acc,t)
chaieb@31120
   533
  in fn t => h ([],t)
chaieb@31120
   534
  end
chaieb@31120
   535
chaieb@31120
   536
 fun f ct =
chaieb@31120
   537
  let 
chaieb@31120
   538
   val nnf_norm_conv' = 
chaieb@31120
   539
     nnf_conv then_conv 
chaieb@31120
   540
     literals_conv [@{term "op &"}, @{term "op |"}] [] 
wenzelm@32843
   541
     (Conv.cache_conv 
chaieb@31120
   542
       (first_conv [real_lt_conv, real_le_conv, 
chaieb@31120
   543
                    real_eq_conv, real_not_lt_conv, 
chaieb@31120
   544
                    real_not_le_conv, real_not_eq_conv, all_conv]))
chaieb@31120
   545
  fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] 
chaieb@31120
   546
                  (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
chaieb@31120
   547
        try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
Philipp@32828
   548
  val nct = Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} ct)
chaieb@31120
   549
  val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
Philipp@32828
   550
  val tm0 = Thm.dest_arg (Thm.rhs_of th0)
Philipp@32645
   551
  val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
chaieb@31120
   552
   let 
chaieb@31120
   553
    val (evs,bod) = strip_exists tm0
chaieb@31120
   554
    val (avs,ibod) = strip_forall bod
chaieb@31120
   555
    val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
Philipp@32828
   556
    val (th2, certs) = overall [] [] [specl avs (assume (Thm.rhs_of th1))]
Philipp@32828
   557
    val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (Thm.capply @{cterm Trueprop} bod))) th2)
Philipp@32645
   558
   in (Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3), certs)
chaieb@31120
   559
   end
Philipp@32645
   560
  in (implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
chaieb@31120
   561
 end
chaieb@31120
   562
in f
chaieb@31120
   563
end;
chaieb@31120
   564
chaieb@31120
   565
(* A linear arithmetic prover *)
chaieb@31120
   566
local
Philipp@32828
   567
  val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
Philipp@32829
   568
  fun linear_cmul c = FuncUtil.Ctermfunc.map (fn x => c */ x)
chaieb@31120
   569
  val one_tm = @{cterm "1::real"}
Philipp@32829
   570
  fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
haftmann@33038
   571
     ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
Philipp@32829
   572
       not(p(FuncUtil.Ctermfunc.apply e one_tm)))
chaieb@31120
   573
chaieb@31120
   574
  fun linear_ineqs vars (les,lts) = 
chaieb@31120
   575
   case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
chaieb@31120
   576
    SOME r => r
chaieb@31120
   577
  | NONE => 
chaieb@31120
   578
   (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
chaieb@31120
   579
     SOME r => r
chaieb@31120
   580
   | NONE => 
chaieb@31120
   581
     if null vars then error "linear_ineqs: no contradiction" else
chaieb@31120
   582
     let 
chaieb@31120
   583
      val ineqs = les @ lts
chaieb@31120
   584
      fun blowup v =
Philipp@32828
   585
       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
Philipp@32828
   586
       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
Philipp@32828
   587
       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
chaieb@31120
   588
      val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
chaieb@31120
   589
                 (map (fn v => (v,blowup v)) vars)))
chaieb@31120
   590
      fun addup (e1,p1) (e2,p2) acc =
chaieb@31120
   591
       let 
Philipp@32828
   592
        val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero 
Philipp@32828
   593
        val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
chaieb@31120
   594
       in if c1 */ c2 >=/ Rat.zero then acc else
chaieb@31120
   595
        let 
chaieb@31120
   596
         val e1' = linear_cmul (Rat.abs c2) e1
chaieb@31120
   597
         val e2' = linear_cmul (Rat.abs c1) e2
chaieb@31120
   598
         val p1' = Product(Rational_lt(Rat.abs c2),p1)
chaieb@31120
   599
         val p2' = Product(Rational_lt(Rat.abs c1),p2)
chaieb@31120
   600
        in (linear_add e1' e2',Sum(p1',p2'))::acc
chaieb@31120
   601
        end
chaieb@31120
   602
       end
chaieb@31120
   603
      val (les0,les1) = 
Philipp@32828
   604
         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
chaieb@31120
   605
      val (lts0,lts1) = 
Philipp@32828
   606
         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
chaieb@31120
   607
      val (lesp,lesn) = 
Philipp@32828
   608
         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
chaieb@31120
   609
      val (ltsp,ltsn) = 
Philipp@32828
   610
         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
chaieb@31120
   611
      val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
chaieb@31120
   612
      val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
chaieb@31120
   613
                      (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
chaieb@31120
   614
     in linear_ineqs (remove (op aconvc) v vars) (les',lts')
chaieb@31120
   615
     end)
chaieb@31120
   616
chaieb@31120
   617
  fun linear_eqs(eqs,les,lts) = 
chaieb@31120
   618
   case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
chaieb@31120
   619
    SOME r => r
chaieb@31120
   620
  | NONE => (case eqs of 
chaieb@31120
   621
    [] => 
chaieb@31120
   622
     let val vars = remove (op aconvc) one_tm 
haftmann@33042
   623
           (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) 
chaieb@31120
   624
     in linear_ineqs vars (les,lts) end
chaieb@31120
   625
   | (e,p)::es => 
Philipp@32829
   626
     if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
chaieb@31120
   627
     let 
Philipp@32829
   628
      val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
chaieb@31120
   629
      fun xform (inp as (t,q)) =
Philipp@32828
   630
       let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
chaieb@31120
   631
        if d =/ Rat.zero then inp else
chaieb@31120
   632
        let 
chaieb@31120
   633
         val k = (Rat.neg d) */ Rat.abs c // c
chaieb@31120
   634
         val e' = linear_cmul k e
chaieb@31120
   635
         val t' = linear_cmul (Rat.abs c) t
Philipp@32829
   636
         val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
chaieb@31120
   637
         val q' = Product(Rational_lt(Rat.abs c),q) 
chaieb@31120
   638
        in (linear_add e' t',Sum(p',q')) 
chaieb@31120
   639
        end 
chaieb@31120
   640
      end
chaieb@31120
   641
     in linear_eqs(map xform es,map xform les,map xform lts)
chaieb@31120
   642
     end)
chaieb@31120
   643
chaieb@31120
   644
  fun linear_prover (eq,le,lt) = 
chaieb@31120
   645
   let 
chaieb@31120
   646
    val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
chaieb@31120
   647
    val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
chaieb@31120
   648
    val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
chaieb@31120
   649
   in linear_eqs(eqs,les,lts)
chaieb@31120
   650
   end 
chaieb@31120
   651
  
chaieb@31120
   652
  fun lin_of_hol ct = 
Philipp@32829
   653
   if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
Philipp@32828
   654
   else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
Philipp@32828
   655
   else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
chaieb@31120
   656
   else
chaieb@31120
   657
    let val (lop,r) = Thm.dest_comb ct 
Philipp@32828
   658
    in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
chaieb@31120
   659
       else
chaieb@31120
   660
        let val (opr,l) = Thm.dest_comb lop 
chaieb@31120
   661
        in if opr aconvc @{cterm "op + :: real =>_"} 
chaieb@31120
   662
           then linear_add (lin_of_hol l) (lin_of_hol r)
chaieb@31120
   663
           else if opr aconvc @{cterm "op * :: real =>_"} 
Philipp@32828
   664
                   andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
Philipp@32828
   665
           else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
chaieb@31120
   666
        end
chaieb@31120
   667
    end
chaieb@31120
   668
chaieb@31120
   669
  fun is_alien ct = case term_of ct of 
chaieb@31120
   670
   Const(@{const_name "real"}, _)$ n => 
chaieb@31120
   671
     if can HOLogic.dest_number n then false else true
chaieb@31120
   672
  | _ => false
chaieb@31120
   673
in 
chaieb@31120
   674
fun real_linear_prover translator (eq,le,lt) = 
chaieb@31120
   675
 let 
Philipp@32828
   676
  val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
Philipp@32828
   677
  val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
chaieb@31120
   678
  val eq_pols = map lhs eq
chaieb@31120
   679
  val le_pols = map rhs le
chaieb@31120
   680
  val lt_pols = map rhs lt 
chaieb@31120
   681
  val aliens =  filter is_alien
haftmann@33042
   682
      (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) 
chaieb@31120
   683
          (eq_pols @ le_pols @ lt_pols) [])
Philipp@32828
   684
  val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
chaieb@31120
   685
  val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
Philipp@32828
   686
  val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
Philipp@32645
   687
 in ((translator (eq,le',lt) proof), Trivial)
chaieb@31120
   688
 end
chaieb@31120
   689
end;
chaieb@31120
   690
chaieb@31120
   691
(* A less general generic arithmetic prover dealing with abs,max and min*)
chaieb@31120
   692
chaieb@31120
   693
local
chaieb@31120
   694
 val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
chaieb@31120
   695
 fun absmaxmin_elim_conv1 ctxt = 
chaieb@31120
   696
    Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
chaieb@31120
   697
chaieb@31120
   698
 val absmaxmin_elim_conv2 =
chaieb@31120
   699
  let 
chaieb@31120
   700
   val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
chaieb@31120
   701
   val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
chaieb@31120
   702
   val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
chaieb@31120
   703
   val abs_tm = @{cterm "abs :: real => _"}
chaieb@31120
   704
   val p_tm = @{cpat "?P :: real => bool"}
chaieb@31120
   705
   val x_tm = @{cpat "?x :: real"}
chaieb@31120
   706
   val y_tm = @{cpat "?y::real"}
chaieb@31120
   707
   val is_max = is_binop @{cterm "max :: real => _"}
chaieb@31120
   708
   val is_min = is_binop @{cterm "min :: real => _"} 
Philipp@32828
   709
   fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
chaieb@31120
   710
   fun eliminate_construct p c tm =
chaieb@31120
   711
    let 
chaieb@31120
   712
     val t = find_cterm p tm
Philipp@32828
   713
     val th0 = (symmetric o beta_conversion false) (Thm.capply (Thm.cabs t tm) t)
Philipp@32828
   714
     val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
chaieb@31120
   715
    in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
chaieb@31120
   716
               (transitive th0 (c p ax))
chaieb@31120
   717
   end
chaieb@31120
   718
chaieb@31120
   719
   val elim_abs = eliminate_construct is_abs
chaieb@31120
   720
    (fn p => fn ax => 
Philipp@32828
   721
       Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
chaieb@31120
   722
   val elim_max = eliminate_construct is_max
chaieb@31120
   723
    (fn p => fn ax => 
Philipp@32828
   724
      let val (ax,y) = Thm.dest_comb ax 
Philipp@32828
   725
      in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
chaieb@31120
   726
      pth_max end)
chaieb@31120
   727
   val elim_min = eliminate_construct is_min
chaieb@31120
   728
    (fn p => fn ax => 
Philipp@32828
   729
      let val (ax,y) = Thm.dest_comb ax 
Philipp@32828
   730
      in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
chaieb@31120
   731
      pth_min end)
chaieb@31120
   732
   in first_conv [elim_abs, elim_max, elim_min, all_conv]
chaieb@31120
   733
  end;
chaieb@31120
   734
in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
chaieb@31120
   735
        gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
chaieb@31120
   736
                       absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
chaieb@31120
   737
end;
chaieb@31120
   738
chaieb@31120
   739
(* An instance for reals*) 
chaieb@31120
   740
chaieb@31120
   741
fun gen_prover_real_arith ctxt prover = 
chaieb@31120
   742
 let
chaieb@31120
   743
  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
chaieb@31120
   744
  val {add,mul,neg,pow,sub,main} = 
chaieb@31120
   745
     Normalizer.semiring_normalizers_ord_wrapper ctxt
wenzelm@33035
   746
      (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
chaieb@31120
   747
     simple_cterm_ord
chaieb@31120
   748
in gen_real_arith ctxt
chaieb@31120
   749
   (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
chaieb@31120
   750
    main,neg,add,mul, prover)
chaieb@31120
   751
end;
chaieb@31120
   752
chaieb@31120
   753
end