--- a/src/HOL/Library/positivstellensatz.ML Wed Feb 22 18:08:41 2012 +0100
+++ b/src/HOL/Library/positivstellensatz.ML Wed Feb 22 19:59:06 2012 +0100
@@ -8,20 +8,20 @@
(* A functor for finite mappings based on Tables *)
-signature FUNC =
+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
+ 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=
+functor FuncFun(Key: KEY) : FUNC =
struct
structure Tab = Table(Key);
@@ -29,24 +29,24 @@
open Tab;
fun dom a = sort Key.ord (Tab.keys a);
-fun applyd f d x = case Tab.lookup f x of
+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;
+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_key f of
- SOME k => (k, the (Tab.lookup f k))
- | NONE => error "FuncFun.choose : Completely empty function"
+fun choose f = case Tab.min_key f of
+ SOME k => (k, the (Tab.lookup f k))
+ | NONE => error "FuncFun.choose : Completely empty function"
fun onefunc kv = update kv empty
@@ -80,94 +80,96 @@
fun dest_monomial mon = sort (cterm_ord o pairself 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
+ 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 cterm_ord int_ord) (mon1,mon2)
- end;
+ 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 cterm_ord int_ord) (mon1,mon2)
+ end;
end
(* positivstellensatz datatype and prover generation *)
-signature REAL_ARITH =
+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;
+ 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 pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
-datatype tree_choice = Left | Right
+ 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
+ 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_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_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 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
+ 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
+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;
+ 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 ->
+type prover = tree_choice list ->
(thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm * pss_tree
+ 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 (cprop_of thb) tha)
+fun deduct_antisym_rule tha thb =
+ Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
(Thm.implies_intr (cprop_of tha) thb);
fun prove_hyp tha thb =
@@ -180,14 +182,14 @@
by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
-val pth_add =
+val pth_add =
@{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
"(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
"(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
"(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
"(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
-val pth_mul =
+val pth_mul =
@{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
"(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
"(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
@@ -273,41 +275,45 @@
(* Miscellaneous *)
-fun literals_conv bops uops cv =
- let fun h t =
- case (term_of t) of
- b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
- | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
- | _ => cv t
- in h end;
+fun literals_conv bops uops cv =
+ let
+ fun h t =
+ case (term_of t) of
+ b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
+ | u$_ => 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.quotient_of_rat x
-in
- if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
- else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
- (Numeral.mk_cnumber @{ctyp "real"} a))
- (Numeral.mk_cnumber @{ctyp "real"} b)
-end;
+fun cterm_of_rat x =
+ let
+ val (a, b) = Rat.quotient_of_rat x
+ in
+ if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
+ else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
+ (Numeral.mk_cnumber @{ctyp "real"} a))
+ (Numeral.mk_cnumber @{ctyp "real"} b)
+ end;
- fun dest_ratconst t = case term_of t of
- Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
- | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
- fun is_ratconst t = can dest_ratconst t
+fun dest_ratconst t =
+ case term_of t of
+ Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+ | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
+fun is_ratconst t = can dest_ratconst t
(*
-fun find_term p t = if p t then t else
+fun find_term p t = if p t then t else
case t of
a$b => (find_term p a handle TERM _ => find_term p b)
| Abs (_,_,t') => find_term p t'
| _ => raise TERM ("find_term",[t]);
*)
-fun find_cterm p t = if p t then t else
- case term_of t of
- _$_ => (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 find_cterm p t =
+ if p t then t else
+ case 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]);
(* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
@@ -319,237 +325,247 @@
(* Map back polynomials to HOL. *)
-fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
+fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
(Numeral.mk_cnumber @{ctyp nat} k)
-fun cterm_of_monomial m =
- if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
- 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 "op * :: real => _"} s) t) vps
- end
+fun cterm_of_monomial m =
+ if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
+ 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 "op * :: real => _"} s) t) vps
+ end
-fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
- else if c = Rat.one then cterm_of_monomial m
- else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
+fun cterm_of_cmonomial (m,c) =
+ if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
+ else if c = Rat.one then cterm_of_monomial m
+ else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
-fun cterm_of_poly p =
- if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
- 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 "op + :: real => _"} t1) t2) cms
- end;
+fun cterm_of_poly p =
+ if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
+ 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 "op + :: real => _"} t1) t2) cms
+ end;
- (* A general real arithmetic prover *)
+(* 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 = HOL_basic_ss 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 = HOL_basic_ss addsimps prenex_simps
- val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
- val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
- val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
- val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
- val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
- val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt 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 "op <= :: real => _"}
- orelse g aconvc @{cterm "op < :: real => _"}
- then arg_conv cv ct else arg1_conv cv ct
- end
+ absconv1,absconv2,prover) =
+ let
+ val pre_ss = HOL_basic_ss 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 = HOL_basic_ss addsimps prenex_simps
+ val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
+ val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
+ val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
+ val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
+ val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
+ val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt 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 "op <= :: real => _"}
+ orelse g aconvc @{cterm "op < :: real => _"}
+ 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))
+ 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))
+ 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))
- (instantiate' [] [SOME ct] (th RS pth_emul))
- fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
+ fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
+ (instantiate' [] [SOME ct] (th RS pth_emul))
+ fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
(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 Trueprop}
- (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x))
+ 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 Trueprop}
+ (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x))
@{cterm "0::real"})))
- | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop}
- (Thm.apply (Thm.apply @{cterm "op <=::real => _"}
+ | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop}
+ (Thm.apply (Thm.apply @{cterm "op <=::real => _"}
@{cterm "0::real"}) (mk_numeric x))))
- | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop}
+ | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop}
(Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"})
(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])
+ | 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 then_conv skolemize_conv then_conv prenex_conv then_conv
- weak_dnf_conv
+ end
+
+ val init_conv = presimp_conv then_conv
+ nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
+ weak_dnf_conv
- val concl = Thm.dest_arg o cprop_of
- fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
- val is_req = is_binop @{cterm "op =:: real => _"}
- val is_ge = is_binop @{cterm "op <=:: real => _"}
- val is_gt = is_binop @{cterm "op <:: real => _"}
- val is_conj = is_binop @{cterm HOL.conj}
- val is_disj = is_binop @{cterm HOL.disj}
- 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
+ val concl = Thm.dest_arg o cprop_of
+ fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
+ val is_req = is_binop @{cterm "op =:: real => _"}
+ val is_ge = is_binop @{cterm "op <=:: real => _"}
+ val is_gt = is_binop @{cterm "op <:: real => _"}
+ val is_conj = is_binop @{cterm HOL.conj}
+ val is_disj = is_binop @{cterm HOL.disj}
+ 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 (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
(Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
(Thm.implies_intr (Thm.apply @{cterm Trueprop} 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 Trueprop} (Thm.dest_arg1 ct))::oths)
- val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (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 "op = :: real => _"}
- 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 ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},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 "uminus :: real => _"} th_p
- val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
- val th' = Drule.binop_cong_rule @{cterm HOL.disj}
- (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
- (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) 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 (term_of t) of
- Const(@{const_name Ex},_)$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 term_of x of
- Free(s,_) => s
- | Var ((s,_),_) => s
- | _ => "x"
+ 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 Trueprop} (Thm.dest_arg1 ct))::oths)
+ val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (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 "op = :: real => _"}
+ 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 ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},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 "uminus :: real => _"} th_p
+ val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
+ val th' = Drule.binop_cong_rule @{cterm HOL.disj}
+ (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
+ (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) 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 (term_of t) of
+ Const(@{const_name Ex},_)$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 term_of x of
+ Free(s,_) => s
+ | Var ((s,_),_) => s
+ | _ => "x"
- fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
+ fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
- val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
+ val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
- fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
- fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
+ fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
+ fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
- fun choose v th th' = case concl_of th of
- @{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
- let
- val p = (funpow 2 Thm.dest_arg o cprop_of) th
- val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
- val th0 = fconv_rule (Thm.beta_conversion true)
- (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
- val pv = (Thm.rhs_of o Thm.beta_conversion true)
- (Thm.apply @{cterm Trueprop} (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 choose v th th' =
+ case concl_of th of
+ @{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
+ let
+ val p = (funpow 2 Thm.dest_arg o cprop_of) th
+ val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
+ val th0 = fconv_rule (Thm.beta_conversion true)
+ (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
+ val pv = (Thm.rhs_of o Thm.beta_conversion true)
+ (Thm.apply @{cterm Trueprop} (Thm.apply p v))
+ val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
+ in Thm.implies_elim (Thm.implies_elim th0 th) th1 end
+ | _ => raise THM ("choose",0,[th, th'])
- fun simple_choose v th =
- choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
+ fun simple_choose v th =
+ choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
- val strip_forall =
- let fun h (acc, t) =
- case (term_of t) of
- Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
- | _ => (acc,t)
- in fn t => h ([],t)
- end
+ val strip_forall =
+ let
+ fun h (acc, t) =
+ case (term_of t) of
+ Const(@{const_name All},_)$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 then_conv
- literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
- (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 HOL.conj}, @{term HOL.disj}] []
- (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 Trueprop} (Thm.apply @{cterm "Not"} 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 False} 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 Trueprop} (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 Trueprop} bod))) th2)
- in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
- end
- in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
- end
-in f
-end;
+ fun f ct =
+ let
+ val nnf_norm_conv' =
+ nnf_conv then_conv
+ literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
+ (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 HOL.conj}, @{term HOL.disj}] []
+ (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 Trueprop} (Thm.apply @{cterm "Not"} 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 False} 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 Trueprop} (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 Trueprop} bod))) th2)
+ in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
+ end
+ in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
+ end
+ in f
+ end;
(* A linear arithmetic prover *)
local
@@ -560,183 +576,190 @@
((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 >/ Rat.zero)) lts of
- SOME r => r
- | NONE =>
- (case find_first (contradictory (fn x => x >/ Rat.zero)) 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 Rat.zero =/ Rat.zero) ineqs) +
- length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
- length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) 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 Rat.zero
- val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
- in if c1 */ c2 >=/ Rat.zero then acc else
- let
- val e1' = linear_cmul (Rat.abs c2) e1
- val e2' = linear_cmul (Rat.abs c1) e2
- val p1' = Product(Rational_lt(Rat.abs c2),p1)
- val p2' = Product(Rational_lt(Rat.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 Rat.zero =/ Rat.zero) les
- val (lts0,lts1) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
- val (lesp,lesn) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
- val (ltsp,ltsn) =
- List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) 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
+ fun linear_ineqs vars (les,lts) =
+ case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
+ SOME r => r
+ | NONE =>
+ (case find_first (contradictory (fn x => x >/ Rat.zero)) 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 Rat.zero =/ Rat.zero) ineqs) +
+ length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
+ length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) 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 Rat.zero
+ val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
+ in
+ if c1 */ c2 >=/ Rat.zero then acc else
+ let
+ val e1' = linear_cmul (Rat.abs c2) e1
+ val e2' = linear_cmul (Rat.abs c1) e2
+ val p1' = Product(Rational_lt(Rat.abs c2),p1)
+ val p2' = Product(Rational_lt(Rat.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 Rat.zero =/ Rat.zero) les
+ val (lts0,lts1) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
+ val (lesp,lesn) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
+ val (ltsp,ltsn) =
+ List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) 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)
+ in linear_ineqs (remove (op aconvc) v vars) (les',lts')
+ end)
- fun linear_eqs(eqs,les,lts) =
- case find_first (contradictory (fn x => x =/ Rat.zero)) 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 Rat.zero in
- if d =/ Rat.zero then inp else
- let
- val k = (Rat.neg d) */ Rat.abs c // c
- val e' = linear_cmul k e
- val t' = linear_cmul (Rat.abs c) t
- val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
- val q' = Product(Rational_lt(Rat.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_eqs(eqs,les,lts) =
+ case find_first (contradictory (fn x => x =/ Rat.zero)) 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 Rat.zero in
+ if d =/ Rat.zero then inp else
+ let
+ val k = (Rat.neg d) */ Rat.abs c // c
+ val e' = linear_cmul k e
+ val t' = linear_cmul (Rat.abs c) t
+ val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
+ val q' = Product(Rational_lt(Rat.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 "0::real"} then FuncUtil.Ctermfunc.empty
- else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
- 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, Rat.one)
- else
- let val (opr,l) = Thm.dest_comb lop
- in if opr aconvc @{cterm "op + :: real =>_"}
- then linear_add (lin_of_hol l) (lin_of_hol r)
- else if opr aconvc @{cterm "op * :: real =>_"}
- andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
- else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
- 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 is_alien ct = case term_of ct of
- Const(@{const_name "real"}, _)$ n =>
- if can HOLogic.dest_number n then false else true
- | _ => 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 cprop_of
- val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o 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,Rat.one)) aliens
- val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
- val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
- in ((translator (eq,le',lt) proof), Trivial)
- end
+ fun lin_of_hol ct =
+ if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
+ else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
+ 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, Rat.one)
+ else
+ let val (opr,l) = Thm.dest_comb lop
+ in
+ if opr aconvc @{cterm "op + :: real =>_"}
+ then linear_add (lin_of_hol l) (lin_of_hol r)
+ else if opr aconvc @{cterm "op * :: real =>_"}
+ andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
+ else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
+ end
+ end
+
+ fun is_alien ct =
+ case term_of ct of
+ Const(@{const_name "real"}, _)$ n =>
+ if can HOLogic.dest_number n then false else true
+ | _ => 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 cprop_of
+ val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o 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,Rat.one)) aliens
+ val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
+ val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) 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 = HOL_basic_ss addsimps real_abs_thms1
- fun absmaxmin_elim_conv1 ctxt =
+ val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
+ fun absmaxmin_elim_conv1 ctxt =
Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
- val absmaxmin_elim_conv2 =
- let
- val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
- val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
- val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
- val abs_tm = @{cterm "abs :: real => _"}
- val p_tm = @{cpat "?P :: real => bool"}
- val x_tm = @{cpat "?x :: real"}
- val y_tm = @{cpat "?y::real"}
- val is_max = is_binop @{cterm "max :: real => _"}
- val is_min = is_binop @{cterm "min :: real => _"}
- 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 absmaxmin_elim_conv2 =
+ let
+ val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
+ val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
+ val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
+ val abs_tm = @{cterm "abs :: real => _"}
+ val p_tm = @{cpat "?P :: real => bool"}
+ val x_tm = @{cpat "?x :: real"}
+ val y_tm = @{cpat "?y::real"}
+ val is_max = is_binop @{cterm "max :: real => _"}
+ val is_min = is_binop @{cterm "min :: real => _"}
+ 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_tm,p), (x_tm, 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_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,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_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,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)
+ val elim_abs = eliminate_construct is_abs
+ (fn p => fn ax =>
+ Thm.instantiate ([], [(p_tm,p), (x_tm, 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_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,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_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,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*)
+(* An instance for reals*)
-fun gen_prover_real_arith ctxt prover =
- let
- fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
- val {add, mul, neg, pow = _, sub = _, main} =
- Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
- simple_cterm_ord
-in gen_real_arith ctxt
- (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
- main,neg,add,mul, prover)
-end;
+fun gen_prover_real_arith ctxt prover =
+ let
+ fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
+ val {add, mul, neg, pow = _, sub = _, main} =
+ Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+ (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+ simple_cterm_ord
+ in gen_real_arith ctxt
+ (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
+ main,neg,add,mul, prover)
+ end;
end