--- a/src/HOL/Library/Sum_Of_Squares/positivstellensatz_tools.ML Sat Jan 08 11:18:09 2011 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,155 +0,0 @@
-(* Title: HOL/Library/Sum_Of_Squares/positivstellensatz_tools.ML
- Author: Philipp Meyer, TU Muenchen
-
-Functions for generating a certificate from a positivstellensatz and vice versa
-*)
-
-signature POSITIVSTELLENSATZ_TOOLS =
-sig
- val pss_tree_to_cert : RealArith.pss_tree -> string
-
- val cert_to_pss_tree : Proof.context -> string -> RealArith.pss_tree
-end
-
-
-structure PositivstellensatzTools : POSITIVSTELLENSATZ_TOOLS =
-struct
-
-(*** certificate generation ***)
-
-fun string_of_rat r =
- let
- val (nom, den) = Rat.quotient_of_rat r
- in
- if den = 1 then string_of_int nom
- else string_of_int nom ^ "/" ^ string_of_int den
- end
-
-(* map polynomials to strings *)
-
-fun string_of_varpow x k =
- let
- val term = term_of x
- val name = case term of
- Free (n, _) => n
- | _ => error "Term in monomial not free variable"
- in
- if k = 1 then name else name ^ "^" ^ string_of_int k
- end
-
-fun string_of_monomial m =
- if FuncUtil.Ctermfunc.is_empty m then "1"
- else
- let
- val m' = FuncUtil.dest_monomial m
- val vps = fold_rev (fn (x,k) => cons (string_of_varpow x k)) m' []
- in foldr1 (fn (s, t) => s ^ "*" ^ t) vps
- end
-
-fun string_of_cmonomial (m,c) =
- if FuncUtil.Ctermfunc.is_empty m then string_of_rat c
- else if c = Rat.one then string_of_monomial m
- else (string_of_rat c) ^ "*" ^ (string_of_monomial m);
-
-fun string_of_poly p =
- if FuncUtil.Monomialfunc.is_empty p then "0"
- else
- let
- val cms = map string_of_cmonomial
- (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
- in foldr1 (fn (t1, t2) => t1 ^ " + " ^ t2) cms
- end;
-
-fun pss_to_cert (RealArith.Axiom_eq i) = "A=" ^ string_of_int i
- | pss_to_cert (RealArith.Axiom_le i) = "A<=" ^ string_of_int i
- | pss_to_cert (RealArith.Axiom_lt i) = "A<" ^ string_of_int i
- | pss_to_cert (RealArith.Rational_eq r) = "R=" ^ string_of_rat r
- | pss_to_cert (RealArith.Rational_le r) = "R<=" ^ string_of_rat r
- | pss_to_cert (RealArith.Rational_lt r) = "R<" ^ string_of_rat r
- | pss_to_cert (RealArith.Square p) = "[" ^ string_of_poly p ^ "]^2"
- | pss_to_cert (RealArith.Eqmul (p, pss)) = "([" ^ string_of_poly p ^ "] * " ^ pss_to_cert pss ^ ")"
- | pss_to_cert (RealArith.Sum (pss1, pss2)) = "(" ^ pss_to_cert pss1 ^ " + " ^ pss_to_cert pss2 ^ ")"
- | pss_to_cert (RealArith.Product (pss1, pss2)) = "(" ^ pss_to_cert pss1 ^ " * " ^ pss_to_cert pss2 ^ ")"
-
-fun pss_tree_to_cert RealArith.Trivial = "()"
- | pss_tree_to_cert (RealArith.Cert pss) = "(" ^ pss_to_cert pss ^ ")"
- | pss_tree_to_cert (RealArith.Branch (t1, t2)) = "(" ^ pss_tree_to_cert t1 ^ " & " ^ pss_tree_to_cert t2 ^ ")"
-
-(*** certificate parsing ***)
-
-(* basic parser *)
-
-val str = Scan.this_string
-
-val number = Scan.repeat1 (Scan.one Symbol.is_ascii_digit >>
- (fn s => ord s - ord "0")) >>
- foldl1 (fn (n, d) => n * 10 + d)
-
-val nat = number
-val int = Scan.optional (str "~" >> K ~1) 1 -- nat >> op *;
-val rat = int --| str "/" -- int >> Rat.rat_of_quotient
-val rat_int = rat || int >> Rat.rat_of_int
-
-(* polynomial parser *)
-
-fun repeat_sep s f = f ::: Scan.repeat (str s |-- f)
-
-val parse_id = Scan.one Symbol.is_letter ::: Scan.many Symbol.is_letdig >> implode
-
-fun parse_varpow ctxt = parse_id -- Scan.optional (str "^" |-- nat) 1 >>
- (fn (x, k) => (cterm_of (ProofContext.theory_of ctxt) (Free (x, @{typ real})), k))
-
-fun parse_monomial ctxt = repeat_sep "*" (parse_varpow ctxt) >>
- (fn xs => fold FuncUtil.Ctermfunc.update xs FuncUtil.Ctermfunc.empty)
-
-fun parse_cmonomial ctxt =
- rat_int --| str "*" -- (parse_monomial ctxt) >> swap ||
- (parse_monomial ctxt) >> (fn m => (m, Rat.one)) ||
- rat_int >> (fn r => (FuncUtil.Ctermfunc.empty, r))
-
-fun parse_poly ctxt = repeat_sep "+" (parse_cmonomial ctxt) >>
- (fn xs => fold FuncUtil.Monomialfunc.update xs FuncUtil.Monomialfunc.empty)
-
-(* positivstellensatz parser *)
-
-val parse_axiom =
- (str "A=" |-- int >> RealArith.Axiom_eq) ||
- (str "A<=" |-- int >> RealArith.Axiom_le) ||
- (str "A<" |-- int >> RealArith.Axiom_lt)
-
-val parse_rational =
- (str "R=" |-- rat_int >> RealArith.Rational_eq) ||
- (str "R<=" |-- rat_int >> RealArith.Rational_le) ||
- (str "R<" |-- rat_int >> RealArith.Rational_lt)
-
-fun parse_cert ctxt input =
- let
- val pc = parse_cert ctxt
- val pp = parse_poly ctxt
- in
- (parse_axiom ||
- parse_rational ||
- str "[" |-- pp --| str "]^2" >> RealArith.Square ||
- str "([" |-- pp --| str "]*" -- pc --| str ")" >> RealArith.Eqmul ||
- str "(" |-- pc --| str "*" -- pc --| str ")" >> RealArith.Product ||
- str "(" |-- pc --| str "+" -- pc --| str ")" >> RealArith.Sum) input
- end
-
-fun parse_cert_tree ctxt input =
- let
- val pc = parse_cert ctxt
- val pt = parse_cert_tree ctxt
- in
- (str "()" >> K RealArith.Trivial ||
- str "(" |-- pc --| str ")" >> RealArith.Cert ||
- str "(" |-- pt --| str "&" -- pt --| str ")" >> RealArith.Branch) input
- end
-
-(* scanner *)
-
-fun cert_to_pss_tree ctxt input_str = Symbol.scanner "bad certificate" (parse_cert_tree ctxt)
- (filter_out Symbol.is_blank (Symbol.explode input_str))
-
-end
-
-