src/HOL/Library/positivstellensatz.ML
author boehmes
Wed Aug 26 11:40:28 2009 +0200 (2009-08-26)
changeset 32402 5731300da417
parent 31971 8c1b845ed105
child 32645 1cc5b24f5a01
permissions -rw-r--r--
added further conversions and conversionals
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(* Title:      Library/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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 type 'a T
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 type key
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 val apply : 'a T -> key -> 'a
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 val applyd :'a T -> (key -> 'a) -> key -> 'a
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 val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
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 val defined : 'a T -> key -> bool
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 val dom : 'a T -> key list
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 val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
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 val fold_rev : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
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 val graph : 'a T -> (key * 'a) list
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 val is_undefined : 'a T -> bool
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 val mapf : ('a -> 'b) -> 'a T -> 'b T
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 val tryapplyd : 'a T -> key -> 'a -> 'a
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 val undefine :  key -> 'a T -> 'a T
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 val undefined : 'a T
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 val update : key * 'a -> 'a T -> 'a T
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 val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
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 val choose : 'a T -> key * 'a
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 val onefunc : key * 'a -> 'a T
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 val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
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end;
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functor FuncFun(Key: KEY) : FUNC=
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struct
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type key = Key.key;
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structure Tab = Table(Key);
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type 'a T = 'a Tab.table;
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val undefined = Tab.empty;
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val is_undefined = Tab.is_empty;
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val mapf = Tab.map;
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val fold = Tab.fold;
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val fold_rev = Tab.fold_rev;
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val graph = Tab.dest;
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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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val defined = Tab.defined;
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fun undefine x t = (Tab.delete x t handle UNDEF => t);
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val update = Tab.update;
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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,valOf (Tab.lookup f k))
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 | NONE => error "FuncFun.choose : Completely undefined function"
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fun onefunc kv = update kv undefined
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local
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fun  find f (k,v) NONE = f (k,v)
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   | find f (k,v) r = r
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in
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fun get_first f t = fold (find f) t NONE
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end
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end;
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structure Intfunc = FuncFun(type key = int val 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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structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t)));
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structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
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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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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 cterm
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 | Eqmul of cterm * positivstellensatz
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 | Sum of positivstellensatz * positivstellensatz
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 | Product of positivstellensatz * positivstellensatz;
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val gen_gen_real_arith :
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  Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv * 
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   conv * conv * conv * conv * conv * conv * 
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    ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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        thm list * thm list * thm list -> thm) -> conv
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val real_linear_prover : 
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  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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   thm list * thm list * thm list -> thm
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val gen_real_arith : Proof.context ->
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   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
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   ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
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       thm list * thm list * thm list -> thm) -> conv
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val gen_prover_real_arith : Proof.context ->
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   ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
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     thm list * thm list * thm list -> thm) -> conv
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val real_arith : Proof.context -> conv
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end
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structure RealArith (* : REAL_ARITH *)=
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struct
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 open Conv Thm;;
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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 cterm
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 | Eqmul of cterm * 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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val my_eqs = ref ([] : thm list);
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val my_les = ref ([] : thm list);
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val my_lts = ref ([] : thm list);
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val my_proof = ref (Axiom_eq 0);
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val my_context = ref @{context};
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val my_mk_numeric = ref ((K @{cterm True}) :Rat.rat -> cterm);
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val my_numeric_eq_conv = ref no_conv;
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val my_numeric_ge_conv = ref no_conv;
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val my_numeric_gt_conv = ref no_conv;
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val my_poly_conv = ref no_conv;
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val my_poly_neg_conv = ref no_conv;
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val my_poly_add_conv = ref no_conv;
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val my_poly_mul_conv = ref no_conv;
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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)) &&&
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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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  by auto};
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val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
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  by (atomize (full)) (auto split add: abs_split)};
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val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
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  by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
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val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
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  by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
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         (* Miscalineous *)
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fun literals_conv bops uops cv = 
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 let fun h t =
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  case (term_of t) of 
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   b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
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 | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
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 | _ => cv t
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 in h end;
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   269
fun cterm_of_rat x = 
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let val (a, b) = Rat.quotient_of_rat x
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in 
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 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
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  else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
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                   (Numeral.mk_cnumber @{ctyp "real"} a))
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        (Numeral.mk_cnumber @{ctyp "real"} b)
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   276
end;
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   278
  fun dest_ratconst t = case term_of t of
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   Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
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 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
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 fun is_ratconst t = can dest_ratconst t
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   282
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   283
fun find_term p t = if p t then t else 
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 case t of
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  a$b => (find_term p a handle TERM _ => find_term p b)
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 | Abs (_,_,t') => find_term p t'
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   287
 | _ => raise TERM ("find_term",[t]);
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   288
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   289
fun find_cterm p t = if p t then t else 
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   290
 case term_of t of
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  a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
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 | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
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   293
 | _ => raise CTERM ("find_cterm",[t]);
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   294
chaieb@31120
   295
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   296
    (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
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fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
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   298
fun is_comb t = case (term_of t) of _$_ => true | _ => false;
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   299
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   300
fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
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  handle CTERM _ => false;
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   302
chaieb@31120
   303
    (* A general real arithmetic prover *)
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   304
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   305
fun gen_gen_real_arith ctxt (mk_numeric,
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       numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
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   307
       poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
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   308
       absconv1,absconv2,prover) = 
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   309
let
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 open Conv Thm;
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 val _ = my_context := ctxt 
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   312
 val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ; 
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          my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
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   314
          my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv; 
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   315
          my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
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 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}]
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 val prenex_ss = HOL_basic_ss addsimps prenex_simps
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 val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
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   319
 val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
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   320
 val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
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   321
 val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
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   322
 val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
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   323
 val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
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   324
 fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
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   325
 fun oprconv cv ct = 
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  let val g = Thm.dest_fun2 ct
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   327
  in if g aconvc @{cterm "op <= :: real => _"} 
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   328
       orelse g aconvc @{cterm "op < :: real => _"} 
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   329
     then arg_conv cv ct else arg1_conv cv ct
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   330
  end
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   331
chaieb@31120
   332
 fun real_ineq_conv th ct =
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  let
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   334
   val th' = (instantiate (match (lhs_of th, ct)) th 
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   335
      handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
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   336
  in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
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   337
  end 
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   338
  val [real_lt_conv, real_le_conv, real_eq_conv,
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   339
       real_not_lt_conv, real_not_le_conv, _] =
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   340
       map real_ineq_conv pth
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   341
  fun match_mp_rule ths ths' = 
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   342
   let
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   343
     fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
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   344
      | th::ths => (ths' MRS th handle THM _ => f ths ths')
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   345
   in f ths ths' end
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   346
  fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
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   347
         (match_mp_rule pth_mul [th, th'])
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   348
  fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
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   349
         (match_mp_rule pth_add [th, th'])
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   350
  fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
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   351
       (instantiate' [] [SOME ct] (th RS pth_emul)) 
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   352
  fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
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   353
       (instantiate' [] [SOME t] pth_square)
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   354
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   355
  fun hol_of_positivstellensatz(eqs,les,lts) proof =
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   356
   let 
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   357
    val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
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   358
    fun translate prf = case prf of
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   359
        Axiom_eq n => nth eqs n
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   360
      | Axiom_le n => nth les n
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   361
      | Axiom_lt n => nth lts n
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   362
      | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} 
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   363
                          (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) 
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   364
                               @{cterm "0::real"})))
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   365
      | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} 
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   366
                          (capply (capply @{cterm "op <=::real => _"} 
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   367
                                     @{cterm "0::real"}) (mk_numeric x))))
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   368
      | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} 
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   369
                      (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
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   370
                        (mk_numeric x))))
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   371
      | Square t => square_rule t
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   372
      | Eqmul(t,p) => emul_rule t (translate p)
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   373
      | Sum(p1,p2) => add_rule (translate p1) (translate p2)
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   374
      | Product(p1,p2) => mul_rule (translate p1) (translate p2)
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   375
   in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
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   376
          (translate proof)
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   377
   end
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   378
  
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   379
  val init_conv = presimp_conv then_conv
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   380
      nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
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   381
      weak_dnf_conv
chaieb@31120
   382
chaieb@31120
   383
  val concl = dest_arg o cprop_of
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   384
  fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
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   385
  val is_req = is_binop @{cterm "op =:: real => _"}
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   386
  val is_ge = is_binop @{cterm "op <=:: real => _"}
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   387
  val is_gt = is_binop @{cterm "op <:: real => _"}
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   388
  val is_conj = is_binop @{cterm "op &"}
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   389
  val is_disj = is_binop @{cterm "op |"}
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   390
  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
chaieb@31120
   391
  fun disj_cases th th1 th2 = 
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   392
   let val (p,q) = dest_binop (concl th)
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   393
       val c = concl th1
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   394
       val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
chaieb@31120
   395
   in implies_elim (implies_elim (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) (implies_intr (capply @{cterm Trueprop} p) th1)) (implies_intr (capply @{cterm Trueprop} q) th2)
chaieb@31120
   396
   end
chaieb@31120
   397
 fun overall dun ths = case ths of
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   398
  [] =>
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   399
   let 
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   400
    val (eq,ne) = List.partition (is_req o concl) dun
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   401
     val (le,nl) = List.partition (is_ge o concl) ne
chaieb@31120
   402
     val lt = filter (is_gt o concl) nl 
chaieb@31120
   403
    in prover hol_of_positivstellensatz (eq,le,lt) end
chaieb@31120
   404
 | th::oths =>
chaieb@31120
   405
   let 
chaieb@31120
   406
    val ct = concl th 
chaieb@31120
   407
   in 
chaieb@31120
   408
    if is_conj ct  then
chaieb@31120
   409
     let 
chaieb@31120
   410
      val (th1,th2) = conj_pair th in
chaieb@31120
   411
      overall dun (th1::th2::oths) end
chaieb@31120
   412
    else if is_disj ct then
chaieb@31120
   413
      let 
chaieb@31120
   414
       val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
chaieb@31120
   415
       val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
chaieb@31120
   416
      in disj_cases th th1 th2 end
chaieb@31120
   417
   else overall (th::dun) oths
chaieb@31120
   418
  end
chaieb@31120
   419
  fun dest_binary b ct = if is_binop b ct then dest_binop ct 
chaieb@31120
   420
                         else raise CTERM ("dest_binary",[b,ct])
chaieb@31120
   421
  val dest_eq = dest_binary @{cterm "op = :: real => _"}
chaieb@31120
   422
  val neq_th = nth pth 5
chaieb@31120
   423
  fun real_not_eq_conv ct = 
chaieb@31120
   424
   let 
chaieb@31120
   425
    val (l,r) = dest_eq (dest_arg ct)
chaieb@31120
   426
    val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
chaieb@31120
   427
    val th_p = poly_conv(dest_arg(dest_arg1(rhs_of th)))
chaieb@31120
   428
    val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
chaieb@31120
   429
    val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
chaieb@31120
   430
    val th' = Drule.binop_cong_rule @{cterm "op |"} 
chaieb@31120
   431
     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
chaieb@31120
   432
     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
chaieb@31120
   433
    in transitive th th' 
chaieb@31120
   434
  end
chaieb@31120
   435
 fun equal_implies_1_rule PQ = 
chaieb@31120
   436
  let 
chaieb@31120
   437
   val P = lhs_of PQ
chaieb@31120
   438
  in implies_intr P (equal_elim PQ (assume P))
chaieb@31120
   439
  end
chaieb@31120
   440
 (* FIXME!!! Copied from groebner.ml *)
chaieb@31120
   441
 val strip_exists =
chaieb@31120
   442
  let fun h (acc, t) =
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   443
   case (term_of t) of
chaieb@31120
   444
    Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
chaieb@31120
   445
  | _ => (acc,t)
chaieb@31120
   446
  in fn t => h ([],t)
chaieb@31120
   447
  end
chaieb@31120
   448
  fun name_of x = case term_of x of
chaieb@31120
   449
   Free(s,_) => s
chaieb@31120
   450
 | Var ((s,_),_) => s
chaieb@31120
   451
 | _ => "x"
chaieb@31120
   452
chaieb@31120
   453
  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
   454
chaieb@31120
   455
  val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
chaieb@31120
   456
chaieb@31120
   457
 fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
chaieb@31120
   458
 fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
chaieb@31120
   459
chaieb@31120
   460
 fun choose v th th' = case concl_of th of 
chaieb@31120
   461
   @{term Trueprop} $ (Const("Ex",_)$_) => 
chaieb@31120
   462
    let
chaieb@31120
   463
     val p = (funpow 2 Thm.dest_arg o cprop_of) th
chaieb@31120
   464
     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
chaieb@31120
   465
     val th0 = fconv_rule (Thm.beta_conversion true)
chaieb@31120
   466
         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
chaieb@31120
   467
     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
chaieb@31120
   468
           (Thm.capply @{cterm Trueprop} (Thm.capply p v))
chaieb@31120
   469
     val th1 = forall_intr v (implies_intr pv th')
chaieb@31120
   470
    in implies_elim (implies_elim th0 th) th1  end
chaieb@31120
   471
 | _ => raise THM ("choose",0,[th, th'])
chaieb@31120
   472
chaieb@31120
   473
  fun simple_choose v th = 
chaieb@31120
   474
     choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
chaieb@31120
   475
chaieb@31120
   476
 val strip_forall =
chaieb@31120
   477
  let fun h (acc, t) =
chaieb@31120
   478
   case (term_of t) of
chaieb@31120
   479
    Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
chaieb@31120
   480
  | _ => (acc,t)
chaieb@31120
   481
  in fn t => h ([],t)
chaieb@31120
   482
  end
chaieb@31120
   483
chaieb@31120
   484
 fun f ct =
chaieb@31120
   485
  let 
chaieb@31120
   486
   val nnf_norm_conv' = 
chaieb@31120
   487
     nnf_conv then_conv 
chaieb@31120
   488
     literals_conv [@{term "op &"}, @{term "op |"}] [] 
boehmes@32402
   489
     (More_Conv.cache_conv 
chaieb@31120
   490
       (first_conv [real_lt_conv, real_le_conv, 
chaieb@31120
   491
                    real_eq_conv, real_not_lt_conv, 
chaieb@31120
   492
                    real_not_le_conv, real_not_eq_conv, all_conv]))
chaieb@31120
   493
  fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] 
chaieb@31120
   494
                  (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
chaieb@31120
   495
        try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
chaieb@31120
   496
  val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
chaieb@31120
   497
  val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
chaieb@31120
   498
  val tm0 = dest_arg (rhs_of th0)
chaieb@31120
   499
  val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else
chaieb@31120
   500
   let 
chaieb@31120
   501
    val (evs,bod) = strip_exists tm0
chaieb@31120
   502
    val (avs,ibod) = strip_forall bod
chaieb@31120
   503
    val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
chaieb@31120
   504
    val th2 = overall [] [specl avs (assume (rhs_of th1))]
chaieb@31120
   505
    val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
chaieb@31120
   506
   in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
chaieb@31120
   507
   end
chaieb@31120
   508
  in implies_elim (instantiate' [] [SOME ct] pth_final) th
chaieb@31120
   509
 end
chaieb@31120
   510
in f
chaieb@31120
   511
end;
chaieb@31120
   512
chaieb@31120
   513
(* A linear arithmetic prover *)
chaieb@31120
   514
local
chaieb@31120
   515
  val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
chaieb@31120
   516
  fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
chaieb@31120
   517
  val one_tm = @{cterm "1::real"}
chaieb@31120
   518
  fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
chaieb@31120
   519
     ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
chaieb@31120
   520
chaieb@31120
   521
  fun linear_ineqs vars (les,lts) = 
chaieb@31120
   522
   case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
chaieb@31120
   523
    SOME r => r
chaieb@31120
   524
  | NONE => 
chaieb@31120
   525
   (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
chaieb@31120
   526
     SOME r => r
chaieb@31120
   527
   | NONE => 
chaieb@31120
   528
     if null vars then error "linear_ineqs: no contradiction" else
chaieb@31120
   529
     let 
chaieb@31120
   530
      val ineqs = les @ lts
chaieb@31120
   531
      fun blowup v =
chaieb@31120
   532
       length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
chaieb@31120
   533
       length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
chaieb@31120
   534
       length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
chaieb@31120
   535
      val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
chaieb@31120
   536
                 (map (fn v => (v,blowup v)) vars)))
chaieb@31120
   537
      fun addup (e1,p1) (e2,p2) acc =
chaieb@31120
   538
       let 
chaieb@31120
   539
        val c1 = Ctermfunc.tryapplyd e1 v Rat.zero 
chaieb@31120
   540
        val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
chaieb@31120
   541
       in if c1 */ c2 >=/ Rat.zero then acc else
chaieb@31120
   542
        let 
chaieb@31120
   543
         val e1' = linear_cmul (Rat.abs c2) e1
chaieb@31120
   544
         val e2' = linear_cmul (Rat.abs c1) e2
chaieb@31120
   545
         val p1' = Product(Rational_lt(Rat.abs c2),p1)
chaieb@31120
   546
         val p2' = Product(Rational_lt(Rat.abs c1),p2)
chaieb@31120
   547
        in (linear_add e1' e2',Sum(p1',p2'))::acc
chaieb@31120
   548
        end
chaieb@31120
   549
       end
chaieb@31120
   550
      val (les0,les1) = 
chaieb@31120
   551
         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
chaieb@31120
   552
      val (lts0,lts1) = 
chaieb@31120
   553
         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
chaieb@31120
   554
      val (lesp,lesn) = 
chaieb@31120
   555
         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
chaieb@31120
   556
      val (ltsp,ltsn) = 
chaieb@31120
   557
         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
chaieb@31120
   558
      val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
chaieb@31120
   559
      val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
chaieb@31120
   560
                      (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
chaieb@31120
   561
     in linear_ineqs (remove (op aconvc) v vars) (les',lts')
chaieb@31120
   562
     end)
chaieb@31120
   563
chaieb@31120
   564
  fun linear_eqs(eqs,les,lts) = 
chaieb@31120
   565
   case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
chaieb@31120
   566
    SOME r => r
chaieb@31120
   567
  | NONE => (case eqs of 
chaieb@31120
   568
    [] => 
chaieb@31120
   569
     let val vars = remove (op aconvc) one_tm 
chaieb@31120
   570
           (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) 
chaieb@31120
   571
     in linear_ineqs vars (les,lts) end
chaieb@31120
   572
   | (e,p)::es => 
chaieb@31120
   573
     if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
chaieb@31120
   574
     let 
chaieb@31120
   575
      val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
chaieb@31120
   576
      fun xform (inp as (t,q)) =
chaieb@31120
   577
       let val d = Ctermfunc.tryapplyd t x Rat.zero in
chaieb@31120
   578
        if d =/ Rat.zero then inp else
chaieb@31120
   579
        let 
chaieb@31120
   580
         val k = (Rat.neg d) */ Rat.abs c // c
chaieb@31120
   581
         val e' = linear_cmul k e
chaieb@31120
   582
         val t' = linear_cmul (Rat.abs c) t
chaieb@31120
   583
         val p' = Eqmul(cterm_of_rat k,p)
chaieb@31120
   584
         val q' = Product(Rational_lt(Rat.abs c),q) 
chaieb@31120
   585
        in (linear_add e' t',Sum(p',q')) 
chaieb@31120
   586
        end 
chaieb@31120
   587
      end
chaieb@31120
   588
     in linear_eqs(map xform es,map xform les,map xform lts)
chaieb@31120
   589
     end)
chaieb@31120
   590
chaieb@31120
   591
  fun linear_prover (eq,le,lt) = 
chaieb@31120
   592
   let 
chaieb@31120
   593
    val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
chaieb@31120
   594
    val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
chaieb@31120
   595
    val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
chaieb@31120
   596
   in linear_eqs(eqs,les,lts)
chaieb@31120
   597
   end 
chaieb@31120
   598
  
chaieb@31120
   599
  fun lin_of_hol ct = 
chaieb@31120
   600
   if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
chaieb@31120
   601
   else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
chaieb@31120
   602
   else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
chaieb@31120
   603
   else
chaieb@31120
   604
    let val (lop,r) = Thm.dest_comb ct 
chaieb@31120
   605
    in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
chaieb@31120
   606
       else
chaieb@31120
   607
        let val (opr,l) = Thm.dest_comb lop 
chaieb@31120
   608
        in if opr aconvc @{cterm "op + :: real =>_"} 
chaieb@31120
   609
           then linear_add (lin_of_hol l) (lin_of_hol r)
chaieb@31120
   610
           else if opr aconvc @{cterm "op * :: real =>_"} 
chaieb@31120
   611
                   andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
chaieb@31120
   612
           else Ctermfunc.onefunc (ct, Rat.one)
chaieb@31120
   613
        end
chaieb@31120
   614
    end
chaieb@31120
   615
chaieb@31120
   616
  fun is_alien ct = case term_of ct of 
chaieb@31120
   617
   Const(@{const_name "real"}, _)$ n => 
chaieb@31120
   618
     if can HOLogic.dest_number n then false else true
chaieb@31120
   619
  | _ => false
chaieb@31120
   620
 open Thm
chaieb@31120
   621
in 
chaieb@31120
   622
fun real_linear_prover translator (eq,le,lt) = 
chaieb@31120
   623
 let 
chaieb@31120
   624
  val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
chaieb@31120
   625
  val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
chaieb@31120
   626
  val eq_pols = map lhs eq
chaieb@31120
   627
  val le_pols = map rhs le
chaieb@31120
   628
  val lt_pols = map rhs lt 
chaieb@31120
   629
  val aliens =  filter is_alien
chaieb@31120
   630
      (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom) 
chaieb@31120
   631
          (eq_pols @ le_pols @ lt_pols) [])
chaieb@31120
   632
  val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
chaieb@31120
   633
  val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
chaieb@31120
   634
  val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
chaieb@31120
   635
 in (translator (eq,le',lt) proof) : thm
chaieb@31120
   636
 end
chaieb@31120
   637
end;
chaieb@31120
   638
chaieb@31120
   639
(* A less general generic arithmetic prover dealing with abs,max and min*)
chaieb@31120
   640
chaieb@31120
   641
local
chaieb@31120
   642
 val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
chaieb@31120
   643
 fun absmaxmin_elim_conv1 ctxt = 
chaieb@31120
   644
    Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
chaieb@31120
   645
chaieb@31120
   646
 val absmaxmin_elim_conv2 =
chaieb@31120
   647
  let 
chaieb@31120
   648
   val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
chaieb@31120
   649
   val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
chaieb@31120
   650
   val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
chaieb@31120
   651
   val abs_tm = @{cterm "abs :: real => _"}
chaieb@31120
   652
   val p_tm = @{cpat "?P :: real => bool"}
chaieb@31120
   653
   val x_tm = @{cpat "?x :: real"}
chaieb@31120
   654
   val y_tm = @{cpat "?y::real"}
chaieb@31120
   655
   val is_max = is_binop @{cterm "max :: real => _"}
chaieb@31120
   656
   val is_min = is_binop @{cterm "min :: real => _"} 
chaieb@31120
   657
   fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
chaieb@31120
   658
   fun eliminate_construct p c tm =
chaieb@31120
   659
    let 
chaieb@31120
   660
     val t = find_cterm p tm
chaieb@31120
   661
     val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
chaieb@31120
   662
     val (p,ax) = (dest_comb o rhs_of) th0
chaieb@31120
   663
    in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
chaieb@31120
   664
               (transitive th0 (c p ax))
chaieb@31120
   665
   end
chaieb@31120
   666
chaieb@31120
   667
   val elim_abs = eliminate_construct is_abs
chaieb@31120
   668
    (fn p => fn ax => 
chaieb@31120
   669
       instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
chaieb@31120
   670
   val elim_max = eliminate_construct is_max
chaieb@31120
   671
    (fn p => fn ax => 
chaieb@31120
   672
      let val (ax,y) = dest_comb ax 
chaieb@31120
   673
      in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
chaieb@31120
   674
      pth_max end)
chaieb@31120
   675
   val elim_min = eliminate_construct is_min
chaieb@31120
   676
    (fn p => fn ax => 
chaieb@31120
   677
      let val (ax,y) = dest_comb ax 
chaieb@31120
   678
      in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
chaieb@31120
   679
      pth_min end)
chaieb@31120
   680
   in first_conv [elim_abs, elim_max, elim_min, all_conv]
chaieb@31120
   681
  end;
chaieb@31120
   682
in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
chaieb@31120
   683
        gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
chaieb@31120
   684
                       absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
chaieb@31120
   685
end;
chaieb@31120
   686
chaieb@31120
   687
(* An instance for reals*) 
chaieb@31120
   688
chaieb@31120
   689
fun gen_prover_real_arith ctxt prover = 
chaieb@31120
   690
 let
chaieb@31120
   691
  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
chaieb@31120
   692
  val {add,mul,neg,pow,sub,main} = 
chaieb@31120
   693
     Normalizer.semiring_normalizers_ord_wrapper ctxt
chaieb@31120
   694
      (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
chaieb@31120
   695
     simple_cterm_ord
chaieb@31120
   696
in gen_real_arith ctxt
chaieb@31120
   697
   (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
chaieb@31120
   698
    main,neg,add,mul, prover)
chaieb@31120
   699
end;
chaieb@31120
   700
chaieb@31120
   701
fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
chaieb@31120
   702
end