src/HOL/Library/Sum_of_Squares/positivstellensatz_tools.ML
 author wenzelm Wed, 01 Jun 2016 15:10:27 +0200 changeset 63205 97b721666890 parent 63201 f151704c08e4 child 63208 3251e9dfea91 permissions -rw-r--r--
prefer rat numberals;
```
(*  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 print_cert: RealArith.pss_tree -> string
val read_cert: Proof.context -> string -> RealArith.pss_tree
end

structure Positivstellensatz_Tools : POSITIVSTELLENSATZ_TOOLS =
struct

(** print certificate **)

local

fun string_of_rat r =
let
val (nom, den) = Rat.dest 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 = Thm.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 = @1 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

(* print cert *)

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 ^ ")"

in

fun print_cert RealArith.Trivial = "()"
| print_cert (RealArith.Cert pss) = "(" ^ pss_to_cert pss ^ ")"
| print_cert (RealArith.Branch (t1, t2)) =
"(" ^ print_cert t1 ^ " & " ^ print_cert t2 ^ ")"

end

local

(* basic parsers *)

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.make
val rat_int = rat || int >> Rat.of_int

(* polynomial parsers *)

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) => (Thm.cterm_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, @1)) ||
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 parsers *)

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

in