--- a/src/HOL/Library/Sum_of_Squares.thy Fri Jun 14 08:34:27 2019 +0000
+++ b/src/HOL/Library/Sum_of_Squares.thy Fri Jun 14 08:34:27 2019 +0000
@@ -10,9 +10,9 @@
imports Complex_Main
begin
-ML_file \<open>positivstellensatz.ML\<close>
+ML_file \<open>Sum_of_Squares/positivstellensatz.ML\<close>
+ML_file \<open>Sum_of_Squares/positivstellensatz_tools.ML\<close>
ML_file \<open>Sum_of_Squares/sum_of_squares.ML\<close>
-ML_file \<open>Sum_of_Squares/positivstellensatz_tools.ML\<close>
ML_file \<open>Sum_of_Squares/sos_wrapper.ML\<close>
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sum_of_Squares/positivstellensatz.ML Fri Jun 14 08:34:27 2019 +0000
@@ -0,0 +1,779 @@
+(* Title: HOL/Library/positivstellensatz.ML
+ Author: Amine Chaieb, University of Cambridge
+
+A generic arithmetic prover based on Positivstellensatz certificates
+--- also implements Fourier-Motzkin elimination as a special case
+Fourier-Motzkin elimination.
+*)
+
+(* A functor for finite mappings based on Tables *)
+
+signature FUNC =
+sig
+ include TABLE
+ val apply : 'a table -> key -> 'a
+ val applyd :'a table -> (key -> 'a) -> key -> 'a
+ val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
+ val dom : 'a table -> key list
+ val tryapplyd : 'a table -> key -> 'a -> 'a
+ val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
+ val choose : 'a table -> key * 'a
+ val onefunc : key * 'a -> 'a table
+end;
+
+functor FuncFun(Key: KEY) : FUNC =
+struct
+
+structure Tab = Table(Key);
+
+open Tab;
+
+fun dom a = sort Key.ord (Tab.keys a);
+fun applyd f d x = case Tab.lookup f x of
+ SOME y => y
+ | NONE => d x;
+
+fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
+fun tryapplyd f a d = applyd f (K d) a;
+fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
+fun combine f z a b =
+ let
+ fun h (k,v) t = case Tab.lookup t k of
+ NONE => Tab.update (k,v) t
+ | SOME v' => let val w = f v v'
+ in if z w then Tab.delete k t else Tab.update (k,w) t end;
+ in Tab.fold h a b end;
+
+fun choose f =
+ (case Tab.min f of
+ SOME entry => entry
+ | NONE => error "FuncFun.choose : Completely empty function")
+
+fun onefunc kv = update kv empty
+
+end;
+
+(* Some standard functors and utility functions for them *)
+
+structure FuncUtil =
+struct
+
+structure Intfunc = FuncFun(type key = int val ord = int_ord);
+structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
+structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
+structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
+structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
+structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);
+
+type monomial = int Ctermfunc.table;
+val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
+structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
+
+type poly = Rat.rat Monomialfunc.table;
+
+(* The ordering so we can create canonical HOL polynomials. *)
+
+fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 fst) (Ctermfunc.dest mon);
+
+fun monomial_order (m1,m2) =
+ if Ctermfunc.is_empty m2 then LESS
+ else if Ctermfunc.is_empty m1 then GREATER
+ else
+ let
+ val mon1 = dest_monomial m1
+ val mon2 = dest_monomial m2
+ val deg1 = fold (Integer.add o snd) mon1 0
+ val deg2 = fold (Integer.add o snd) mon2 0
+ in if deg1 < deg2 then GREATER
+ else if deg1 > deg2 then LESS
+ else list_ord (prod_ord Thm.fast_term_ord int_ord) (mon1,mon2)
+ end;
+
+end
+
+(* positivstellensatz datatype and prover generation *)
+
+signature REAL_ARITH =
+sig
+
+ datatype positivstellensatz =
+ Axiom_eq of int
+ | Axiom_le of int
+ | Axiom_lt of int
+ | Rational_eq of Rat.rat
+ | Rational_le of Rat.rat
+ | Rational_lt of Rat.rat
+ | Square of FuncUtil.poly
+ | Eqmul of FuncUtil.poly * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz;
+
+ datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
+
+ datatype tree_choice = Left | Right
+
+ type prover = tree_choice list ->
+ (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+ thm list * thm list * thm list -> thm * pss_tree
+ type cert_conv = cterm -> thm * pss_tree
+
+ val gen_gen_real_arith :
+ Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
+ conv * conv * conv * conv * conv * conv * prover -> cert_conv
+ val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+ thm list * thm list * thm list -> thm * pss_tree
+
+ val gen_real_arith : Proof.context ->
+ (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
+
+ val gen_prover_real_arith : Proof.context -> prover -> cert_conv
+
+ val is_ratconst : cterm -> bool
+ val dest_ratconst : cterm -> Rat.rat
+ val cterm_of_rat : Rat.rat -> cterm
+
+end
+
+structure RealArith : REAL_ARITH =
+struct
+
+open Conv
+(* ------------------------------------------------------------------------- *)
+(* Data structure for Positivstellensatz refutations. *)
+(* ------------------------------------------------------------------------- *)
+
+datatype positivstellensatz =
+ Axiom_eq of int
+ | Axiom_le of int
+ | Axiom_lt of int
+ | Rational_eq of Rat.rat
+ | Rational_le of Rat.rat
+ | Rational_lt of Rat.rat
+ | Square of FuncUtil.poly
+ | Eqmul of FuncUtil.poly * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz;
+ (* Theorems used in the procedure *)
+
+datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
+datatype tree_choice = Left | Right
+type prover = tree_choice list ->
+ (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+ thm list * thm list * thm list -> thm * pss_tree
+type cert_conv = cterm -> thm * pss_tree
+
+
+ (* Some useful derived rules *)
+fun deduct_antisym_rule tha thb =
+ Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha)
+ (Thm.implies_intr (Thm.cprop_of tha) thb);
+
+fun prove_hyp tha thb =
+ if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb) (* FIXME !? *)
+ then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
+
+val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and
+ "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and
+ "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))"
+ by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
+
+val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast}
+val pth_add =
+ @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and
+ "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and
+ "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and
+ "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and
+ "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all};
+
+val pth_mul =
+ @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and
+ "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and
+ "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and
+ "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and
+ "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y > 0)"
+ by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
+ mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
+
+val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0" by simp};
+val pth_square = @{lemma "x * x \<ge> (0::real)" by simp};
+
+val weak_dnf_simps =
+ List.take (@{thms simp_thms}, 34) @
+ @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and
+ "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+};
+
+(*
+val nnfD_simps =
+ @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
+ "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
+ "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
+*)
+
+val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>x. P x (f x))" by metis};
+val prenex_simps =
+ map (fn th => th RS sym)
+ ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
+ @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
+
+val real_abs_thms1 = @{lemma
+ "((-1 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and
+ "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
+ "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
+ "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and
+ "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and
+ "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and
+ "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and
+ "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
+ "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
+ "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y + b \<ge> r))" and
+ "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and
+ "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y + c \<ge> r))" and
+ "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and
+ "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
+ "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
+ "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y + b \<ge> r))" and
+ "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and
+ "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y + c \<ge> r))" and
+ "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and
+ "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
+ "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
+ "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y + b \<ge> r))" and
+ "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and
+ "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and
+ "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and
+ "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
+ "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
+ "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and
+ "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and
+ "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and
+ "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and
+ "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
+ "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
+ "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y + b > r))" and
+ "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and
+ "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y + c > r))" and
+ "((min x y > r) = (x > r \<and> y > r))" and
+ "((min x y + a > r) = (a + x > r \<and> a + y > r))" and
+ "((a + min x y > r) = (a + x > r \<and> a + y > r))" and
+ "((a + min x y + b > r) = (a + x + b > r \<and> a + y + b > r))" and
+ "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and
+ "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))"
+ by auto};
+
+val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))"
+ by (atomize (full)) (auto split: abs_split)};
+
+val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)"
+ by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)};
+
+val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)"
+ by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)};
+
+
+ (* Miscellaneous *)
+fun literals_conv bops uops cv =
+ let
+ fun h t =
+ (case Thm.term_of t of
+ b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
+ | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
+ | _ => cv t)
+ in h end;
+
+fun cterm_of_rat x =
+ let
+ val (a, b) = Rat.dest x
+ in
+ if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a
+ else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
+ (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a))
+ (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b)
+ end;
+
+fun dest_ratconst t =
+ case Thm.term_of t of
+ Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+ | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
+fun is_ratconst t = can dest_ratconst t
+
+(*
+fun find_term p t = if p t then t else
+ case t of
+ a$b => (find_term p a handle TERM _ => find_term p b)
+ | Abs (_,_,t') => find_term p t'
+ | _ => raise TERM ("find_term",[t]);
+*)
+
+fun find_cterm p t =
+ if p t then t else
+ case Thm.term_of t of
+ _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
+ | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
+ | _ => raise CTERM ("find_cterm",[t]);
+
+fun is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false);
+
+fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
+ handle CTERM _ => false;
+
+
+(* Map back polynomials to HOL. *)
+
+fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close> x)
+ (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k)
+
+fun cterm_of_monomial m =
+ if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close>
+ else
+ let
+ val m' = FuncUtil.dest_monomial m
+ val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
+ in foldr1 (fn (s, t) => Thm.apply (Thm.apply \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> s) t) vps
+ end
+
+fun cterm_of_cmonomial (m,c) =
+ if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
+ else if c = @1 then cterm_of_monomial m
+ else Thm.apply (Thm.apply \<^cterm>\<open>(*)::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m);
+
+fun cterm_of_poly p =
+ if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close>
+ else
+ let
+ val cms = map cterm_of_cmonomial
+ (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
+ in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close> t1) t2) cms
+ end;
+
+(* A general real arithmetic prover *)
+
+fun gen_gen_real_arith ctxt (mk_numeric,
+ numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
+ poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
+ absconv1,absconv2,prover) =
+ let
+ val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
+ @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib
+ all_conj_distrib if_bool_eq_disj}
+ val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
+ val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
+ val presimp_conv = Simplifier.rewrite pre_ss
+ val prenex_conv = Simplifier.rewrite prenex_ss
+ val skolemize_conv = Simplifier.rewrite skolemize_ss
+ val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
+ val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss
+ fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
+ fun oprconv cv ct =
+ let val g = Thm.dest_fun2 ct
+ in if g aconvc \<^cterm>\<open>(\<le>) :: real \<Rightarrow> _\<close>
+ orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
+ then arg_conv cv ct else arg1_conv cv ct
+ end
+
+ fun real_ineq_conv th ct =
+ let
+ val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
+ handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
+ in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
+ end
+ val [real_lt_conv, real_le_conv, real_eq_conv,
+ real_not_lt_conv, real_not_le_conv, _] =
+ map real_ineq_conv pth
+ fun match_mp_rule ths ths' =
+ let
+ fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
+ | th::ths => (ths' MRS th handle THM _ => f ths ths')
+ in f ths ths' end
+ fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
+ (match_mp_rule pth_mul [th, th'])
+ fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
+ (match_mp_rule pth_add [th, th'])
+ fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
+ (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
+ fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
+ (Thm.instantiate' [] [SOME t] pth_square)
+
+ fun hol_of_positivstellensatz(eqs,les,lts) proof =
+ let
+ fun translate prf =
+ case prf of
+ Axiom_eq n => nth eqs n
+ | Axiom_le n => nth les n
+ | Axiom_lt n => nth lts n
+ | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
+ (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x))
+ \<^cterm>\<open>0::real\<close>)))
+ | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
+ (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close>
+ \<^cterm>\<open>0::real\<close>) (mk_numeric x))))
+ | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
+ (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>)
+ (mk_numeric x))))
+ | Square pt => square_rule (cterm_of_poly pt)
+ | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
+ | Sum(p1,p2) => add_rule (translate p1) (translate p2)
+ | Product(p1,p2) => mul_rule (translate p1) (translate p2)
+ in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
+ (translate proof)
+ end
+
+ val init_conv = presimp_conv then_conv
+ nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
+ weak_dnf_conv
+
+ val concl = Thm.dest_arg o Thm.cprop_of
+ fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
+ val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close>
+ val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close>
+ val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close>
+ val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close>
+ val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close>
+ fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
+ fun disj_cases th th1 th2 =
+ let
+ val (p,q) = Thm.dest_binop (concl th)
+ val c = concl th1
+ val _ =
+ if c aconvc (concl th2) then ()
+ else error "disj_cases : conclusions not alpha convertible"
+ in Thm.implies_elim (Thm.implies_elim
+ (Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
+ (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1))
+ (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> q) th2)
+ end
+ fun overall cert_choice dun ths =
+ case ths of
+ [] =>
+ let
+ val (eq,ne) = List.partition (is_req o concl) dun
+ val (le,nl) = List.partition (is_ge o concl) ne
+ val lt = filter (is_gt o concl) nl
+ in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
+ | th::oths =>
+ let
+ val ct = concl th
+ in
+ if is_conj ct then
+ let
+ val (th1,th2) = conj_pair th
+ in overall cert_choice dun (th1::th2::oths) end
+ else if is_disj ct then
+ let
+ val (th1, cert1) =
+ overall (Left::cert_choice) dun
+ (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths)
+ val (th2, cert2) =
+ overall (Right::cert_choice) dun
+ (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg ct))::oths)
+ in (disj_cases th th1 th2, Branch (cert1, cert2)) end
+ else overall cert_choice (th::dun) oths
+ end
+ fun dest_binary b ct =
+ if is_binop b ct then Thm.dest_binop ct
+ else raise CTERM ("dest_binary",[b,ct])
+ val dest_eq = dest_binary \<^cterm>\<open>(=) :: real \<Rightarrow> _\<close>
+ val neq_th = nth pth 5
+ fun real_not_eq_conv ct =
+ let
+ val (l,r) = dest_eq (Thm.dest_arg ct)
+ val th = Thm.instantiate ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),r)]) neq_th
+ val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
+ val th_x = Drule.arg_cong_rule \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close> th_p
+ val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
+ val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close>
+ (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p)
+ (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_n)
+ in Thm.transitive th th'
+ end
+ fun equal_implies_1_rule PQ =
+ let
+ val P = Thm.lhs_of PQ
+ in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
+ end
+ (*FIXME!!! Copied from groebner.ml*)
+ val strip_exists =
+ let
+ fun h (acc, t) =
+ case Thm.term_of t of
+ Const(\<^const_name>\<open>Ex\<close>,_)$Abs(_,_,_) =>
+ h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
+ | _ => (acc,t)
+ in fn t => h ([],t)
+ end
+ fun name_of x =
+ case Thm.term_of x of
+ Free(s,_) => s
+ | Var ((s,_),_) => s
+ | _ => "x"
+
+ fun mk_forall x th =
+ let
+ val T = Thm.typ_of_cterm x
+ val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
+ in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
+
+ val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
+
+ fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
+ fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
+
+ fun choose v th th' =
+ case Thm.concl_of th of
+ \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) =>
+ let
+ val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
+ val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p)
+ val th0 = fconv_rule (Thm.beta_conversion true)
+ (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
+ val pv = (Thm.rhs_of o Thm.beta_conversion true)
+ (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v))
+ val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
+ in Thm.implies_elim (Thm.implies_elim th0 th) th1 end
+ | _ => raise THM ("choose",0,[th, th'])
+
+ fun simple_choose v th =
+ choose v
+ (Thm.assume
+ ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
+
+ val strip_forall =
+ let
+ fun h (acc, t) =
+ case Thm.term_of t of
+ Const(\<^const_name>\<open>All\<close>,_)$Abs(_,_,_) =>
+ h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
+ | _ => (acc,t)
+ in fn t => h ([],t)
+ end
+
+ fun f ct =
+ let
+ val nnf_norm_conv' =
+ nnf_conv ctxt then_conv
+ literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
+ (Conv.cache_conv
+ (first_conv [real_lt_conv, real_le_conv,
+ real_eq_conv, real_not_lt_conv,
+ real_not_le_conv, real_not_eq_conv, all_conv]))
+ fun absremover ct = (literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
+ (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
+ try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
+ val nct = Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> ct)
+ val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
+ val tm0 = Thm.dest_arg (Thm.rhs_of th0)
+ val (th, certificates) =
+ if tm0 aconvc \<^cterm>\<open>False\<close> then (equal_implies_1_rule th0, Trivial) else
+ let
+ val (evs,bod) = strip_exists tm0
+ val (avs,ibod) = strip_forall bod
+ val th1 = Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (fold mk_forall avs (absremover ibod))
+ val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
+ val th3 =
+ fold simple_choose evs
+ (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> bod))) th2)
+ in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
+ end
+ in (Thm.implies_elim (Thm.instantiate' [] [SOME ct] pth_final) th, certificates)
+ end
+ in f
+ end;
+
+(* A linear arithmetic prover *)
+local
+ val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = @0)
+ fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
+ val one_tm = \<^cterm>\<open>1::real\<close>
+ fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) orelse
+ ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
+ not(p(FuncUtil.Ctermfunc.apply e one_tm)))
+
+ fun linear_ineqs vars (les,lts) =
+ case find_first (contradictory (fn x => x > @0)) lts of
+ SOME r => r
+ | NONE =>
+ (case find_first (contradictory (fn x => x > @0)) les of
+ SOME r => r
+ | NONE =>
+ if null vars then error "linear_ineqs: no contradiction" else
+ let
+ val ineqs = les @ lts
+ fun blowup v =
+ length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) ineqs) +
+ length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
+ length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) ineqs)
+ val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
+ (map (fn v => (v,blowup v)) vars)))
+ fun addup (e1,p1) (e2,p2) acc =
+ let
+ val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v @0
+ val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
+ in
+ if c1 * c2 >= @0 then acc else
+ let
+ val e1' = linear_cmul (abs c2) e1
+ val e2' = linear_cmul (abs c1) e2
+ val p1' = Product(Rational_lt (abs c2), p1)
+ val p2' = Product(Rational_lt (abs c1), p2)
+ in (linear_add e1' e2',Sum(p1',p2'))::acc
+ end
+ end
+ val (les0,les1) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) les
+ val (lts0,lts1) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
+ val (lesp,lesn) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
+ val (ltsp,ltsn) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) lts1
+ val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
+ val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
+ (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
+ in linear_ineqs (remove (op aconvc) v vars) (les',lts')
+ end)
+
+ fun linear_eqs(eqs,les,lts) =
+ case find_first (contradictory (fn x => x = @0)) eqs of
+ SOME r => r
+ | NONE =>
+ (case eqs of
+ [] =>
+ let val vars = remove (op aconvc) one_tm
+ (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
+ in linear_ineqs vars (les,lts) end
+ | (e,p)::es =>
+ if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
+ let
+ val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
+ fun xform (inp as (t,q)) =
+ let val d = FuncUtil.Ctermfunc.tryapplyd t x @0 in
+ if d = @0 then inp else
+ let
+ val k = ~ d * abs c / c
+ val e' = linear_cmul k e
+ val t' = linear_cmul (abs c) t
+ val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
+ val q' = Product(Rational_lt (abs c), q)
+ in (linear_add e' t',Sum(p',q'))
+ end
+ end
+ in linear_eqs(map xform es,map xform les,map xform lts)
+ end)
+
+ fun linear_prover (eq,le,lt) =
+ let
+ val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
+ val les = map_index (fn (n, p) => (p,Axiom_le n)) le
+ val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
+ in linear_eqs(eqs,les,lts)
+ end
+
+ fun lin_of_hol ct =
+ if ct aconvc \<^cterm>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty
+ else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
+ else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
+ else
+ let val (lop,r) = Thm.dest_comb ct
+ in
+ if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, @1)
+ else
+ let val (opr,l) = Thm.dest_comb lop
+ in
+ if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
+ then linear_add (lin_of_hol l) (lin_of_hol r)
+ else if opr aconvc \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
+ andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
+ else FuncUtil.Ctermfunc.onefunc (ct, @1)
+ end
+ end
+
+ fun is_alien ct =
+ case Thm.term_of ct of
+ Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n)
+ | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n)
+ | _ => false
+in
+fun real_linear_prover translator (eq,le,lt) =
+ let
+ val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of
+ val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of
+ val eq_pols = map lhs eq
+ val le_pols = map rhs le
+ val lt_pols = map rhs lt
+ val aliens = filter is_alien
+ (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
+ (eq_pols @ le_pols @ lt_pols) [])
+ val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,@1)) aliens
+ val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
+ val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
+ in ((translator (eq,le',lt) proof), Trivial)
+ end
+end;
+
+(* A less general generic arithmetic prover dealing with abs,max and min*)
+
+local
+ val absmaxmin_elim_ss1 =
+ simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1)
+ fun absmaxmin_elim_conv1 ctxt =
+ Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
+
+ val absmaxmin_elim_conv2 =
+ let
+ val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split'
+ val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split
+ val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split
+ val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close>
+ val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>)
+ val x_v = (("x", 0), \<^typ>\<open>real\<close>)
+ val y_v = (("y", 0), \<^typ>\<open>real\<close>)
+ val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close>
+ val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close>
+ fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
+ fun eliminate_construct p c tm =
+ let
+ val t = find_cterm p tm
+ val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
+ val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
+ in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
+ (Thm.transitive th0 (c p ax))
+ end
+
+ val elim_abs = eliminate_construct is_abs
+ (fn p => fn ax =>
+ Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax)]) pth_abs)
+ val elim_max = eliminate_construct is_max
+ (fn p => fn ax =>
+ let val (ax,y) = Thm.dest_comb ax
+ in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
+ pth_max end)
+ val elim_min = eliminate_construct is_min
+ (fn p => fn ax =>
+ let val (ax,y) = Thm.dest_comb ax
+ in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
+ pth_min end)
+ in first_conv [elim_abs, elim_max, elim_min, all_conv]
+ end;
+in
+fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
+ gen_gen_real_arith ctxt
+ (mkconst,eq,ge,gt,norm,neg,add,mul,
+ absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
+end;
+
+(* An instance for reals*)
+
+fun gen_prover_real_arith ctxt prover =
+ let
+ val {add, mul, neg, pow = _, sub = _, main} =
+ Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+ (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
+ Thm.term_ord
+ in gen_real_arith ctxt
+ (cterm_of_rat,
+ Numeral_Simprocs.field_comp_conv ctxt,
+ Numeral_Simprocs.field_comp_conv ctxt,
+ Numeral_Simprocs.field_comp_conv ctxt,
+ main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
+ end;
+
+end
--- a/src/HOL/Library/positivstellensatz.ML Fri Jun 14 08:34:27 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,779 +0,0 @@
-(* Title: HOL/Library/positivstellensatz.ML
- Author: Amine Chaieb, University of Cambridge
-
-A generic arithmetic prover based on Positivstellensatz certificates
---- also implements Fourier-Motzkin elimination as a special case
-Fourier-Motzkin elimination.
-*)
-
-(* A functor for finite mappings based on Tables *)
-
-signature FUNC =
-sig
- include TABLE
- val apply : 'a table -> key -> 'a
- val applyd :'a table -> (key -> 'a) -> key -> 'a
- val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
- val dom : 'a table -> key list
- val tryapplyd : 'a table -> key -> 'a -> 'a
- val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
- val choose : 'a table -> key * 'a
- val onefunc : key * 'a -> 'a table
-end;
-
-functor FuncFun(Key: KEY) : FUNC =
-struct
-
-structure Tab = Table(Key);
-
-open Tab;
-
-fun dom a = sort Key.ord (Tab.keys a);
-fun applyd f d x = case Tab.lookup f x of
- SOME y => y
- | NONE => d x;
-
-fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
-fun tryapplyd f a d = applyd f (K d) a;
-fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
-fun combine f z a b =
- let
- fun h (k,v) t = case Tab.lookup t k of
- NONE => Tab.update (k,v) t
- | SOME v' => let val w = f v v'
- in if z w then Tab.delete k t else Tab.update (k,w) t end;
- in Tab.fold h a b end;
-
-fun choose f =
- (case Tab.min f of
- SOME entry => entry
- | NONE => error "FuncFun.choose : Completely empty function")
-
-fun onefunc kv = update kv empty
-
-end;
-
-(* Some standard functors and utility functions for them *)
-
-structure FuncUtil =
-struct
-
-structure Intfunc = FuncFun(type key = int val ord = int_ord);
-structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
-structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
-structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
-structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
-structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);
-
-type monomial = int Ctermfunc.table;
-val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
-structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
-
-type poly = Rat.rat Monomialfunc.table;
-
-(* The ordering so we can create canonical HOL polynomials. *)
-
-fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 fst) (Ctermfunc.dest mon);
-
-fun monomial_order (m1,m2) =
- if Ctermfunc.is_empty m2 then LESS
- else if Ctermfunc.is_empty m1 then GREATER
- else
- let
- val mon1 = dest_monomial m1
- val mon2 = dest_monomial m2
- val deg1 = fold (Integer.add o snd) mon1 0
- val deg2 = fold (Integer.add o snd) mon2 0
- in if deg1 < deg2 then GREATER
- else if deg1 > deg2 then LESS
- else list_ord (prod_ord Thm.fast_term_ord int_ord) (mon1,mon2)
- end;
-
-end
-
-(* positivstellensatz datatype and prover generation *)
-
-signature REAL_ARITH =
-sig
-
- datatype positivstellensatz =
- Axiom_eq of int
- | Axiom_le of int
- | Axiom_lt of int
- | Rational_eq of Rat.rat
- | Rational_le of Rat.rat
- | Rational_lt of Rat.rat
- | Square of FuncUtil.poly
- | Eqmul of FuncUtil.poly * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
-
- datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
-
- datatype tree_choice = Left | Right
-
- type prover = tree_choice list ->
- (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm * pss_tree
- type cert_conv = cterm -> thm * pss_tree
-
- val gen_gen_real_arith :
- Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
- conv * conv * conv * conv * conv * conv * prover -> cert_conv
- val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm * pss_tree
-
- val gen_real_arith : Proof.context ->
- (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
-
- val gen_prover_real_arith : Proof.context -> prover -> cert_conv
-
- val is_ratconst : cterm -> bool
- val dest_ratconst : cterm -> Rat.rat
- val cterm_of_rat : Rat.rat -> cterm
-
-end
-
-structure RealArith : REAL_ARITH =
-struct
-
-open Conv
-(* ------------------------------------------------------------------------- *)
-(* Data structure for Positivstellensatz refutations. *)
-(* ------------------------------------------------------------------------- *)
-
-datatype positivstellensatz =
- Axiom_eq of int
- | Axiom_le of int
- | Axiom_lt of int
- | Rational_eq of Rat.rat
- | Rational_le of Rat.rat
- | Rational_lt of Rat.rat
- | Square of FuncUtil.poly
- | Eqmul of FuncUtil.poly * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
- (* Theorems used in the procedure *)
-
-datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
-datatype tree_choice = Left | Right
-type prover = tree_choice list ->
- (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm * pss_tree
-type cert_conv = cterm -> thm * pss_tree
-
-
- (* Some useful derived rules *)
-fun deduct_antisym_rule tha thb =
- Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha)
- (Thm.implies_intr (Thm.cprop_of tha) thb);
-
-fun prove_hyp tha thb =
- if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb) (* FIXME !? *)
- then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
-
-val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and
- "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and
- "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))"
- by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
-
-val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast}
-val pth_add =
- @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and
- "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and
- "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and
- "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and
- "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all};
-
-val pth_mul =
- @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and
- "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and
- "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and
- "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and
- "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y > 0)"
- by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
- mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
-
-val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0" by simp};
-val pth_square = @{lemma "x * x \<ge> (0::real)" by simp};
-
-val weak_dnf_simps =
- List.take (@{thms simp_thms}, 34) @
- @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and
- "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+};
-
-(*
-val nnfD_simps =
- @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
- "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
- "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
-*)
-
-val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>x. P x (f x))" by metis};
-val prenex_simps =
- map (fn th => th RS sym)
- ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
- @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
-
-val real_abs_thms1 = @{lemma
- "((-1 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and
- "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
- "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
- "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and
- "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and
- "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and
- "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and
- "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
- "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
- "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y + b \<ge> r))" and
- "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and
- "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y + c \<ge> r))" and
- "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and
- "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
- "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
- "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y + b \<ge> r))" and
- "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and
- "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y + c \<ge> r))" and
- "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and
- "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
- "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
- "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y + b \<ge> r))" and
- "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and
- "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and
- "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and
- "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
- "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
- "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and
- "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and
- "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and
- "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and
- "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
- "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
- "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y + b > r))" and
- "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and
- "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y + c > r))" and
- "((min x y > r) = (x > r \<and> y > r))" and
- "((min x y + a > r) = (a + x > r \<and> a + y > r))" and
- "((a + min x y > r) = (a + x > r \<and> a + y > r))" and
- "((a + min x y + b > r) = (a + x + b > r \<and> a + y + b > r))" and
- "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and
- "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))"
- by auto};
-
-val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))"
- by (atomize (full)) (auto split: abs_split)};
-
-val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)"
- by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)};
-
-val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)"
- by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)};
-
-
- (* Miscellaneous *)
-fun literals_conv bops uops cv =
- let
- fun h t =
- (case Thm.term_of t of
- b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
- | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
- | _ => cv t)
- in h end;
-
-fun cterm_of_rat x =
- let
- val (a, b) = Rat.dest x
- in
- if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a
- else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
- (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a))
- (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b)
- end;
-
-fun dest_ratconst t =
- case Thm.term_of t of
- Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
- | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
-fun is_ratconst t = can dest_ratconst t
-
-(*
-fun find_term p t = if p t then t else
- case t of
- a$b => (find_term p a handle TERM _ => find_term p b)
- | Abs (_,_,t') => find_term p t'
- | _ => raise TERM ("find_term",[t]);
-*)
-
-fun find_cterm p t =
- if p t then t else
- case Thm.term_of t of
- _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
- | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
- | _ => raise CTERM ("find_cterm",[t]);
-
-fun is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false);
-
-fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
- handle CTERM _ => false;
-
-
-(* Map back polynomials to HOL. *)
-
-fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close> x)
- (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k)
-
-fun cterm_of_monomial m =
- if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close>
- else
- let
- val m' = FuncUtil.dest_monomial m
- val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
- in foldr1 (fn (s, t) => Thm.apply (Thm.apply \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> s) t) vps
- end
-
-fun cterm_of_cmonomial (m,c) =
- if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
- else if c = @1 then cterm_of_monomial m
- else Thm.apply (Thm.apply \<^cterm>\<open>(*)::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m);
-
-fun cterm_of_poly p =
- if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close>
- else
- let
- val cms = map cterm_of_cmonomial
- (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
- in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close> t1) t2) cms
- end;
-
-(* A general real arithmetic prover *)
-
-fun gen_gen_real_arith ctxt (mk_numeric,
- numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
- poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
- absconv1,absconv2,prover) =
- let
- val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
- @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib
- all_conj_distrib if_bool_eq_disj}
- val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
- val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
- val presimp_conv = Simplifier.rewrite pre_ss
- val prenex_conv = Simplifier.rewrite prenex_ss
- val skolemize_conv = Simplifier.rewrite skolemize_ss
- val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
- val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss
- fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
- fun oprconv cv ct =
- let val g = Thm.dest_fun2 ct
- in if g aconvc \<^cterm>\<open>(\<le>) :: real \<Rightarrow> _\<close>
- orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
- then arg_conv cv ct else arg1_conv cv ct
- end
-
- fun real_ineq_conv th ct =
- let
- val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
- handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
- in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
- end
- val [real_lt_conv, real_le_conv, real_eq_conv,
- real_not_lt_conv, real_not_le_conv, _] =
- map real_ineq_conv pth
- fun match_mp_rule ths ths' =
- let
- fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
- | th::ths => (ths' MRS th handle THM _ => f ths ths')
- in f ths ths' end
- fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
- (match_mp_rule pth_mul [th, th'])
- fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
- (match_mp_rule pth_add [th, th'])
- fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
- (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
- fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
- (Thm.instantiate' [] [SOME t] pth_square)
-
- fun hol_of_positivstellensatz(eqs,les,lts) proof =
- let
- fun translate prf =
- case prf of
- Axiom_eq n => nth eqs n
- | Axiom_le n => nth les n
- | Axiom_lt n => nth lts n
- | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
- (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x))
- \<^cterm>\<open>0::real\<close>)))
- | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
- (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close>
- \<^cterm>\<open>0::real\<close>) (mk_numeric x))))
- | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
- (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>)
- (mk_numeric x))))
- | Square pt => square_rule (cterm_of_poly pt)
- | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
- | Sum(p1,p2) => add_rule (translate p1) (translate p2)
- | Product(p1,p2) => mul_rule (translate p1) (translate p2)
- in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
- (translate proof)
- end
-
- val init_conv = presimp_conv then_conv
- nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
- weak_dnf_conv
-
- val concl = Thm.dest_arg o Thm.cprop_of
- fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
- val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close>
- val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close>
- val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close>
- val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close>
- val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close>
- fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
- fun disj_cases th th1 th2 =
- let
- val (p,q) = Thm.dest_binop (concl th)
- val c = concl th1
- val _ =
- if c aconvc (concl th2) then ()
- else error "disj_cases : conclusions not alpha convertible"
- in Thm.implies_elim (Thm.implies_elim
- (Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
- (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1))
- (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> q) th2)
- end
- fun overall cert_choice dun ths =
- case ths of
- [] =>
- let
- val (eq,ne) = List.partition (is_req o concl) dun
- val (le,nl) = List.partition (is_ge o concl) ne
- val lt = filter (is_gt o concl) nl
- in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
- | th::oths =>
- let
- val ct = concl th
- in
- if is_conj ct then
- let
- val (th1,th2) = conj_pair th
- in overall cert_choice dun (th1::th2::oths) end
- else if is_disj ct then
- let
- val (th1, cert1) =
- overall (Left::cert_choice) dun
- (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths)
- val (th2, cert2) =
- overall (Right::cert_choice) dun
- (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg ct))::oths)
- in (disj_cases th th1 th2, Branch (cert1, cert2)) end
- else overall cert_choice (th::dun) oths
- end
- fun dest_binary b ct =
- if is_binop b ct then Thm.dest_binop ct
- else raise CTERM ("dest_binary",[b,ct])
- val dest_eq = dest_binary \<^cterm>\<open>(=) :: real \<Rightarrow> _\<close>
- val neq_th = nth pth 5
- fun real_not_eq_conv ct =
- let
- val (l,r) = dest_eq (Thm.dest_arg ct)
- val th = Thm.instantiate ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),r)]) neq_th
- val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
- val th_x = Drule.arg_cong_rule \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close> th_p
- val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
- val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close>
- (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p)
- (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_n)
- in Thm.transitive th th'
- end
- fun equal_implies_1_rule PQ =
- let
- val P = Thm.lhs_of PQ
- in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
- end
- (*FIXME!!! Copied from groebner.ml*)
- val strip_exists =
- let
- fun h (acc, t) =
- case Thm.term_of t of
- Const(\<^const_name>\<open>Ex\<close>,_)$Abs(_,_,_) =>
- h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
- | _ => (acc,t)
- in fn t => h ([],t)
- end
- fun name_of x =
- case Thm.term_of x of
- Free(s,_) => s
- | Var ((s,_),_) => s
- | _ => "x"
-
- fun mk_forall x th =
- let
- val T = Thm.typ_of_cterm x
- val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
- in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
-
- val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
-
- fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
- fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
-
- fun choose v th th' =
- case Thm.concl_of th of
- \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) =>
- let
- val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
- val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p)
- val th0 = fconv_rule (Thm.beta_conversion true)
- (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
- val pv = (Thm.rhs_of o Thm.beta_conversion true)
- (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v))
- val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
- in Thm.implies_elim (Thm.implies_elim th0 th) th1 end
- | _ => raise THM ("choose",0,[th, th'])
-
- fun simple_choose v th =
- choose v
- (Thm.assume
- ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
-
- val strip_forall =
- let
- fun h (acc, t) =
- case Thm.term_of t of
- Const(\<^const_name>\<open>All\<close>,_)$Abs(_,_,_) =>
- h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
- | _ => (acc,t)
- in fn t => h ([],t)
- end
-
- fun f ct =
- let
- val nnf_norm_conv' =
- nnf_conv ctxt then_conv
- literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
- (Conv.cache_conv
- (first_conv [real_lt_conv, real_le_conv,
- real_eq_conv, real_not_lt_conv,
- real_not_le_conv, real_not_eq_conv, all_conv]))
- fun absremover ct = (literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
- (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
- try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
- val nct = Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> ct)
- val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
- val tm0 = Thm.dest_arg (Thm.rhs_of th0)
- val (th, certificates) =
- if tm0 aconvc \<^cterm>\<open>False\<close> then (equal_implies_1_rule th0, Trivial) else
- let
- val (evs,bod) = strip_exists tm0
- val (avs,ibod) = strip_forall bod
- val th1 = Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (fold mk_forall avs (absremover ibod))
- val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
- val th3 =
- fold simple_choose evs
- (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> bod))) th2)
- in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
- end
- in (Thm.implies_elim (Thm.instantiate' [] [SOME ct] pth_final) th, certificates)
- end
- in f
- end;
-
-(* A linear arithmetic prover *)
-local
- val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = @0)
- fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
- val one_tm = \<^cterm>\<open>1::real\<close>
- fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) orelse
- ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
- not(p(FuncUtil.Ctermfunc.apply e one_tm)))
-
- fun linear_ineqs vars (les,lts) =
- case find_first (contradictory (fn x => x > @0)) lts of
- SOME r => r
- | NONE =>
- (case find_first (contradictory (fn x => x > @0)) les of
- SOME r => r
- | NONE =>
- if null vars then error "linear_ineqs: no contradiction" else
- let
- val ineqs = les @ lts
- fun blowup v =
- length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) ineqs) +
- length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
- length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) ineqs)
- val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
- (map (fn v => (v,blowup v)) vars)))
- fun addup (e1,p1) (e2,p2) acc =
- let
- val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v @0
- val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
- in
- if c1 * c2 >= @0 then acc else
- let
- val e1' = linear_cmul (abs c2) e1
- val e2' = linear_cmul (abs c1) e2
- val p1' = Product(Rational_lt (abs c2), p1)
- val p2' = Product(Rational_lt (abs c1), p2)
- in (linear_add e1' e2',Sum(p1',p2'))::acc
- end
- end
- val (les0,les1) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) les
- val (lts0,lts1) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
- val (lesp,lesn) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
- val (ltsp,ltsn) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) lts1
- val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
- val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
- (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
- in linear_ineqs (remove (op aconvc) v vars) (les',lts')
- end)
-
- fun linear_eqs(eqs,les,lts) =
- case find_first (contradictory (fn x => x = @0)) eqs of
- SOME r => r
- | NONE =>
- (case eqs of
- [] =>
- let val vars = remove (op aconvc) one_tm
- (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
- in linear_ineqs vars (les,lts) end
- | (e,p)::es =>
- if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
- let
- val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
- fun xform (inp as (t,q)) =
- let val d = FuncUtil.Ctermfunc.tryapplyd t x @0 in
- if d = @0 then inp else
- let
- val k = ~ d * abs c / c
- val e' = linear_cmul k e
- val t' = linear_cmul (abs c) t
- val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
- val q' = Product(Rational_lt (abs c), q)
- in (linear_add e' t',Sum(p',q'))
- end
- end
- in linear_eqs(map xform es,map xform les,map xform lts)
- end)
-
- fun linear_prover (eq,le,lt) =
- let
- val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
- val les = map_index (fn (n, p) => (p,Axiom_le n)) le
- val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
- in linear_eqs(eqs,les,lts)
- end
-
- fun lin_of_hol ct =
- if ct aconvc \<^cterm>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty
- else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
- else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
- else
- let val (lop,r) = Thm.dest_comb ct
- in
- if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, @1)
- else
- let val (opr,l) = Thm.dest_comb lop
- in
- if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
- then linear_add (lin_of_hol l) (lin_of_hol r)
- else if opr aconvc \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
- andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
- else FuncUtil.Ctermfunc.onefunc (ct, @1)
- end
- end
-
- fun is_alien ct =
- case Thm.term_of ct of
- Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n)
- | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n)
- | _ => false
-in
-fun real_linear_prover translator (eq,le,lt) =
- let
- val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of
- val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of
- val eq_pols = map lhs eq
- val le_pols = map rhs le
- val lt_pols = map rhs lt
- val aliens = filter is_alien
- (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
- (eq_pols @ le_pols @ lt_pols) [])
- val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,@1)) aliens
- val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
- val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
- in ((translator (eq,le',lt) proof), Trivial)
- end
-end;
-
-(* A less general generic arithmetic prover dealing with abs,max and min*)
-
-local
- val absmaxmin_elim_ss1 =
- simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1)
- fun absmaxmin_elim_conv1 ctxt =
- Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
-
- val absmaxmin_elim_conv2 =
- let
- val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split'
- val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split
- val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split
- val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close>
- val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>)
- val x_v = (("x", 0), \<^typ>\<open>real\<close>)
- val y_v = (("y", 0), \<^typ>\<open>real\<close>)
- val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close>
- val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close>
- fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
- fun eliminate_construct p c tm =
- let
- val t = find_cterm p tm
- val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
- val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
- in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
- (Thm.transitive th0 (c p ax))
- end
-
- val elim_abs = eliminate_construct is_abs
- (fn p => fn ax =>
- Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax)]) pth_abs)
- val elim_max = eliminate_construct is_max
- (fn p => fn ax =>
- let val (ax,y) = Thm.dest_comb ax
- in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
- pth_max end)
- val elim_min = eliminate_construct is_min
- (fn p => fn ax =>
- let val (ax,y) = Thm.dest_comb ax
- in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
- pth_min end)
- in first_conv [elim_abs, elim_max, elim_min, all_conv]
- end;
-in
-fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
- gen_gen_real_arith ctxt
- (mkconst,eq,ge,gt,norm,neg,add,mul,
- absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
-end;
-
-(* An instance for reals*)
-
-fun gen_prover_real_arith ctxt prover =
- let
- val {add, mul, neg, pow = _, sub = _, main} =
- Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
- Thm.term_ord
- in gen_real_arith ctxt
- (cterm_of_rat,
- Numeral_Simprocs.field_comp_conv ctxt,
- Numeral_Simprocs.field_comp_conv ctxt,
- Numeral_Simprocs.field_comp_conv ctxt,
- main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
- end;
-
-end