author | Christian Sternagel |
Thu, 30 Aug 2012 15:44:03 +0900 | |
changeset 49093 | fdc301f592c4 |
parent 43946 | ba88bb44c192 |
child 55508 | 90c42b130652 |
permissions | -rw-r--r-- |
41474 | 1 |
(* Title: HOL/Library/Sum_of_Squares/positivstellensatz_tools.ML |
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Author: Philipp Meyer, TU Muenchen |
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Functions for generating a certificate from a positivstellensatz and vice |
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versa. |
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*) |
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signature POSITIVSTELLENSATZ_TOOLS = |
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sig |
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val pss_tree_to_cert : RealArith.pss_tree -> string |
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val cert_to_pss_tree : Proof.context -> string -> RealArith.pss_tree |
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end |
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structure PositivstellensatzTools : POSITIVSTELLENSATZ_TOOLS = |
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struct |
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(*** certificate generation ***) |
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fun string_of_rat r = |
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let |
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val (nom, den) = Rat.quotient_of_rat r |
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in |
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if den = 1 then string_of_int nom |
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else string_of_int nom ^ "/" ^ string_of_int den |
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end |
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(* map polynomials to strings *) |
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fun string_of_varpow x k = |
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let |
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val term = term_of x |
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val name = case term of |
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Free (n, _) => n |
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| _ => error "Term in monomial not free variable" |
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in |
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if k = 1 then name else name ^ "^" ^ string_of_int k |
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end |
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fun string_of_monomial m = |
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671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset
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if FuncUtil.Ctermfunc.is_empty m then "1" |
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else |
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let |
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val m' = FuncUtil.dest_monomial m |
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val vps = fold_rev (fn (x,k) => cons (string_of_varpow x k)) m' [] |
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in foldr1 (fn (s, t) => s ^ "*" ^ t) vps |
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end |
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fun string_of_cmonomial (m,c) = |
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32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset
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if FuncUtil.Ctermfunc.is_empty m then string_of_rat c |
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else if c = Rat.one then string_of_monomial m |
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else (string_of_rat c) ^ "*" ^ (string_of_monomial m); |
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fun string_of_poly p = |
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32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset
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if FuncUtil.Monomialfunc.is_empty p then "0" |
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else |
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let |
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val cms = map string_of_cmonomial |
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32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset
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(sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p)) |
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in foldr1 (fn (t1, t2) => t1 ^ " + " ^ t2) cms |
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end; |
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fun pss_to_cert (RealArith.Axiom_eq i) = "A=" ^ string_of_int i |
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| pss_to_cert (RealArith.Axiom_le i) = "A<=" ^ string_of_int i |
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| pss_to_cert (RealArith.Axiom_lt i) = "A<" ^ string_of_int i |
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| pss_to_cert (RealArith.Rational_eq r) = "R=" ^ string_of_rat r |
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| pss_to_cert (RealArith.Rational_le r) = "R<=" ^ string_of_rat r |
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| pss_to_cert (RealArith.Rational_lt r) = "R<" ^ string_of_rat r |
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| pss_to_cert (RealArith.Square p) = "[" ^ string_of_poly p ^ "]^2" |
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| pss_to_cert (RealArith.Eqmul (p, pss)) = "([" ^ string_of_poly p ^ "] * " ^ pss_to_cert pss ^ ")" |
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| pss_to_cert (RealArith.Sum (pss1, pss2)) = "(" ^ pss_to_cert pss1 ^ " + " ^ pss_to_cert pss2 ^ ")" |
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| pss_to_cert (RealArith.Product (pss1, pss2)) = "(" ^ pss_to_cert pss1 ^ " * " ^ pss_to_cert pss2 ^ ")" |
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fun pss_tree_to_cert RealArith.Trivial = "()" |
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| pss_tree_to_cert (RealArith.Cert pss) = "(" ^ pss_to_cert pss ^ ")" |
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| pss_tree_to_cert (RealArith.Branch (t1, t2)) = "(" ^ pss_tree_to_cert t1 ^ " & " ^ pss_tree_to_cert t2 ^ ")" |
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(*** certificate parsing ***) |
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(* basic parser *) |
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val str = Scan.this_string |
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val number = Scan.repeat1 (Scan.one Symbol.is_ascii_digit >> |
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(fn s => ord s - ord "0")) >> |
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foldl1 (fn (n, d) => n * 10 + d) |
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val nat = number |
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val int = Scan.optional (str "~" >> K ~1) 1 -- nat >> op *; |
962b4354ed90
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parents:
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changeset
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val rat = int --| str "/" -- int >> Rat.rat_of_quotient |
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val rat_int = rat || int >> Rat.rat_of_int |
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(* polynomial parser *) |
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parents:
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diff
changeset
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fun repeat_sep s f = f ::: Scan.repeat (str s |-- f) |
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val parse_id = Scan.one Symbol.is_letter ::: Scan.many Symbol.is_letdig >> implode |
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962b4354ed90
used standard fold function and type aliases
Philipp Meyer
parents:
32645
diff
changeset
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fun parse_varpow ctxt = parse_id -- Scan.optional (str "^" |-- nat) 1 >> |
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(fn (x, k) => (cterm_of (Proof_Context.theory_of ctxt) (Free (x, @{typ real})), k)) |
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fun parse_monomial ctxt = repeat_sep "*" (parse_varpow ctxt) >> |
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(fn xs => fold FuncUtil.Ctermfunc.update xs FuncUtil.Ctermfunc.empty) |
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fun parse_cmonomial ctxt = |
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962b4354ed90
used standard fold function and type aliases
Philipp Meyer
parents:
32645
diff
changeset
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rat_int --| str "*" -- (parse_monomial ctxt) >> swap || |
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(parse_monomial ctxt) >> (fn m => (m, Rat.one)) || |
32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset
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rat_int >> (fn r => (FuncUtil.Ctermfunc.empty, r)) |
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fun parse_poly ctxt = repeat_sep "+" (parse_cmonomial ctxt) >> |
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(fn xs => fold FuncUtil.Monomialfunc.update xs FuncUtil.Monomialfunc.empty) |
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(* positivstellensatz parser *) |
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val parse_axiom = |
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(str "A=" |-- int >> RealArith.Axiom_eq) || |
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(str "A<=" |-- int >> RealArith.Axiom_le) || |
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(str "A<" |-- int >> RealArith.Axiom_lt) |
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val parse_rational = |
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(str "R=" |-- rat_int >> RealArith.Rational_eq) || |
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(str "R<=" |-- rat_int >> RealArith.Rational_le) || |
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(str "R<" |-- rat_int >> RealArith.Rational_lt) |
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fun parse_cert ctxt input = |
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let |
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val pc = parse_cert ctxt |
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val pp = parse_poly ctxt |
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in |
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(parse_axiom || |
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parse_rational || |
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str "[" |-- pp --| str "]^2" >> RealArith.Square || |
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str "([" |-- pp --| str "]*" -- pc --| str ")" >> RealArith.Eqmul || |
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str "(" |-- pc --| str "*" -- pc --| str ")" >> RealArith.Product || |
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str "(" |-- pc --| str "+" -- pc --| str ")" >> RealArith.Sum) input |
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end |
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fun parse_cert_tree ctxt input = |
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let |
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val pc = parse_cert ctxt |
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val pt = parse_cert_tree ctxt |
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in |
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(str "()" >> K RealArith.Trivial || |
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str "(" |-- pc --| str ")" >> RealArith.Cert || |
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str "(" |-- pt --| str "&" -- pt --| str ")" >> RealArith.Branch) input |
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end |
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(* scanner *) |
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fun cert_to_pss_tree ctxt input_str = |
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Symbol.scanner "Bad certificate" (parse_cert_tree ctxt) |
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(filter_out Symbol.is_blank (Symbol.explode input_str)) |
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end |
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