src/HOL/Library/Sum_of_Squares/positivstellensatz_tools.ML
 changeset 41474 60d091240485 parent 37744 3daaf23b9ab4 child 42361 23f352990944
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sum_of_Squares/positivstellensatz_tools.ML	Sat Jan 08 17:39:51 2011 +0100
@@ -0,0 +1,156 @@
+(*  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
+
+```