--- a/src/HOL/Library/Sum_of_Squares/sum_of_squares.ML Mon Aug 22 10:43:10 2011 -0700
+++ b/src/HOL/Library/Sum_of_Squares/sum_of_squares.ML Mon Aug 22 16:49:45 2011 -0700
@@ -20,12 +20,18 @@
val rat_1 = Rat.one;
val rat_2 = Rat.two;
val rat_10 = Rat.rat_of_int 10;
+(*
val rat_1_2 = rat_1 // rat_2;
+*)
val max = Integer.max;
+(*
val min = Integer.min;
+*)
val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
+(*
val numerator_rat = Rat.quotient_of_rat #> fst #> Rat.rat_of_int;
+*)
fun int_of_rat a =
case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
fun lcm_rat x y = Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
@@ -99,7 +105,7 @@
type matrix = (int*int)*(Rat.rat FuncUtil.Intpairfunc.table);
-fun iszero (k,r) = r =/ rat_0;
+fun iszero (_,r) = r =/ rat_0;
(* Vectors. Conventionally indexed 1..n. *)
@@ -108,11 +114,13 @@
fun dim (v:vector) = fst v;
+(*
fun vector_const c n =
if c =/ rat_0 then vector_0 n
else (n,fold_rev (fn k => FuncUtil.Intfunc.update (k,c)) (1 upto n) FuncUtil.Intfunc.empty) :vector;
val vector_1 = vector_const rat_1;
+*)
fun vector_cmul c (v:vector) =
let val n = dim v
@@ -120,6 +128,7 @@
else (n,FuncUtil.Intfunc.map (fn _ => fn x => c */ x) (snd v))
end;
+(*
fun vector_neg (v:vector) = (fst v,FuncUtil.Intfunc.map (K Rat.neg) (snd v)) :vector;
fun vector_add (v1:vector) (v2:vector) =
@@ -135,9 +144,10 @@
let val m = dim v1
val n = dim v2
in if m <> n then error "vector_dot: incompatible dimensions"
- else FuncUtil.Intfunc.fold (fn (i,x) => fn a => x +/ a)
+ else FuncUtil.Intfunc.fold (fn (_,x) => fn a => x +/ a)
(FuncUtil.Intfunc.combine (curry op */) (fn x => x =/ rat_0) (snd v1) (snd v2)) rat_0
end;
+*)
fun vector_of_list l =
let val n = length l
@@ -146,10 +156,13 @@
(* Matrices; again rows and columns indexed from 1. *)
+(*
fun matrix_0 (m,n) = ((m,n),FuncUtil.Intpairfunc.empty):matrix;
+*)
fun dimensions (m:matrix) = fst m;
+(*
fun matrix_const c (mn as (m,n)) =
if m <> n then error "matrix_const: needs to be square"
else if c =/ rat_0 then matrix_0 mn
@@ -175,15 +188,17 @@
end;;
fun matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);
+*)
fun row k (m:matrix) =
- let val (i,j) = dimensions m
+ let val (_,j) = dimensions m
in (j,
FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if i = k then FuncUtil.Intfunc.update (j,c) a else a) (snd m) FuncUtil.Intfunc.empty ) : vector
end;
+(*
fun column k (m:matrix) =
- let val (i,j) = dimensions m
+ let val (i,_) = dimensions m
in (i,
FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if j = k then FuncUtil.Intfunc.update (i,c) a else a) (snd m) FuncUtil.Intfunc.empty)
: vector
@@ -207,6 +222,7 @@
in ((m,n),itern 1 l (fn v => fn i => itern 1 v (fn c => fn j => FuncUtil.Intpairfunc.update ((i,j), c))) FuncUtil.Intpairfunc.empty)
end
end;
+*)
(* Monomials. *)
@@ -220,6 +236,7 @@
val monomial_mul =
FuncUtil.Ctermfunc.combine Integer.add (K false);
+(*
fun monomial_pow m k =
if k = 0 then monomial_1
else FuncUtil.Ctermfunc.map (fn _ => fn x => k * x) m;
@@ -230,7 +247,7 @@
fun monomial_div m1 m2 =
let val m = FuncUtil.Ctermfunc.combine Integer.add
(fn x => x = 0) m1 (FuncUtil.Ctermfunc.map (fn _ => fn x => ~ x) m2)
- in if FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => k >= 0 andalso a) m true then m
+ in if FuncUtil.Ctermfunc.fold (fn (_, k) => fn a => k >= 0 andalso a) m true then m
else error "monomial_div: non-divisible"
end;
@@ -240,9 +257,10 @@
fun monomial_lcm m1 m2 =
fold_rev (fn x => FuncUtil.Ctermfunc.update (x, max (monomial_degree x m1) (monomial_degree x m2)))
(union (is_equal o FuncUtil.cterm_ord) (FuncUtil.Ctermfunc.dom m1) (FuncUtil.Ctermfunc.dom m2)) (FuncUtil.Ctermfunc.empty);
+*)
fun monomial_multidegree m =
- FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => k + a) m 0;;
+ FuncUtil.Ctermfunc.fold (fn (_, k) => fn a => k + a) m 0;;
fun monomial_variables m = FuncUtil.Ctermfunc.dom m;;
@@ -254,7 +272,7 @@
val poly_0 = FuncUtil.Monomialfunc.empty;
fun poly_isconst p =
- FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a) p true;
+ FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a) p true;
fun poly_var x = FuncUtil.Monomialfunc.onefunc (monomial_var x,rat_1);
@@ -281,6 +299,7 @@
fun poly_mul p1 p2 =
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
+(*
fun poly_div p1 p2 =
if not(poly_isconst p2)
then error "poly_div: non-constant" else
@@ -288,6 +307,7 @@
in if c =/ rat_0 then error "poly_div: division by zero"
else poly_cmul (Rat.inv c) p1
end;
+*)
fun poly_square p = poly_mul p p;
@@ -297,22 +317,25 @@
else let val q = poly_square(poly_pow p (k div 2)) in
if k mod 2 = 1 then poly_mul p q else q end;
+(*
fun poly_exp p1 p2 =
if not(poly_isconst p2)
then error "poly_exp: not a constant"
else poly_pow p1 (int_of_rat (eval FuncUtil.Ctermfunc.empty p2));
fun degree x p =
- FuncUtil.Monomialfunc.fold (fn (m,c) => fn a => max (monomial_degree x m) a) p 0;
+ FuncUtil.Monomialfunc.fold (fn (m,_) => fn a => max (monomial_degree x m) a) p 0;
+*)
fun multidegree p =
- FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => max (monomial_multidegree m) a) p 0;
+ FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => max (monomial_multidegree m) a) p 0;
fun poly_variables p =
- sort FuncUtil.cterm_ord (FuncUtil.Monomialfunc.fold_rev (fn (m, c) => union (is_equal o FuncUtil.cterm_ord) (monomial_variables m)) p []);;
+ sort FuncUtil.cterm_ord (FuncUtil.Monomialfunc.fold_rev (fn (m, _) => union (is_equal o FuncUtil.cterm_ord) (monomial_variables m)) p []);;
(* Order monomials for human presentation. *)
+(*
val humanorder_varpow = prod_ord FuncUtil.cterm_ord (rev_order o int_ord);
local
@@ -328,9 +351,11 @@
ord (sort humanorder_varpow (FuncUtil.Ctermfunc.dest m1),
sort humanorder_varpow (FuncUtil.Ctermfunc.dest m2))
end;
+*)
(* Conversions to strings. *)
+(*
fun string_of_vector min_size max_size (v:vector) =
let val n_raw = dim v
in if n_raw = 0 then "[]" else
@@ -394,6 +419,7 @@
val s2 = String.substring (s, 3, String.size s - 3)
in "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>"
end;
+*)
(* Conversion from HOL term. *)
@@ -407,7 +433,9 @@
val pow_tm = @{cterm "op ^ :: real => _"}
val zero_tm = @{cterm "0:: real"}
val is_numeral = can (HOLogic.dest_number o term_of)
+(*
fun is_comb t = case t of _$_ => true | _ => false
+*)
fun poly_of_term tm =
if tm aconvc zero_tm then poly_0
else if RealArith.is_ratconst tm
@@ -466,6 +494,7 @@
(* String for block diagonal matrix numbered k. *)
+(*
fun sdpa_of_blockdiagonal k m =
let
val pfx = string_of_int k ^" "
@@ -476,9 +505,11 @@
pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
" " ^ decimalize 20 c ^ "\n" ^ a) entss ""
end;
+*)
(* String for a matrix numbered k, in SDPA sparse format. *)
+(*
fun sdpa_of_matrix k (m:matrix) =
let
val pfx = string_of_int k ^ " 1 "
@@ -488,6 +519,7 @@
pfx ^ string_of_int i ^ " " ^ string_of_int j ^
" " ^ decimalize 20 c ^ "\n" ^ a) mss ""
end;;
+*)
(* ------------------------------------------------------------------------- *)
(* String in SDPA sparse format for standard SDP problem: *)
@@ -496,6 +528,7 @@
(* Minimize obj_1 * v_1 + ... obj_m * v_m *)
(* ------------------------------------------------------------------------- *)
+(*
fun sdpa_of_problem obj mats =
let
val m = length mats - 1
@@ -507,6 +540,7 @@
sdpa_of_vector obj ^
fold_rev2 (fn k => fn m => fn a => sdpa_of_matrix (k - 1) m ^ a) (1 upto length mats) mats ""
end;
+*)
fun index_char str chr pos =
if pos >= String.size str then ~1
@@ -523,14 +557,18 @@
end
end;
+(*
fun isspace x = (x = " ");
+*)
fun isnum x = member (op =) ["0","1","2","3","4","5","6","7","8","9"] x;
(* More parser basics. *)
+(*
val word = Scan.this_string
fun token s =
Scan.repeat ($$ " ") |-- word s --| Scan.repeat ($$ " ")
+*)
val numeral = Scan.one isnum
val decimalint = Scan.repeat1 numeral >> (rat_of_string o implode)
val decimalfrac = Scan.repeat1 numeral
@@ -558,7 +596,7 @@
(* Parse back csdp output. *)
- fun ignore inp = ((),[])
+ fun ignore _ = ((),[])
fun csdpoutput inp =
((decimal -- Scan.repeat (Scan.$$ " " |-- Scan.option decimal) >>
(fn (h,to) => map_filter I ((SOME h)::to))) --| ignore >> vector_of_list) inp
@@ -566,8 +604,10 @@
(* Run prover on a problem in linear form. *)
+(*
fun run_problem prover obj mats =
parse_csdpoutput (prover (sdpa_of_problem obj mats))
+*)
(* Try some apparently sensible scaling first. Note that this is purely to *)
(* get a cleaner translation to floating-point, and doesn't affect any of *)
@@ -575,6 +615,7 @@
(* are extreme numbers in the original problem. *)
(* Version for (int*int) keys *)
+(*
local
fun max_rat x y = if x </ y then y else x
fun common_denominator fld amat acc =
@@ -600,6 +641,7 @@
in solver obj'' mats''
end
end;
+*)
(* Try some apparently sensible scaling first. Note that this is purely to *)
(* get a cleaner translation to floating-point, and doesn't affect any of *)
@@ -610,9 +652,9 @@
local
fun max_rat x y = if x </ y then y else x
fun common_denominator fld amat acc =
- fld (fn (m,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
+ fld (fn (_,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
fun maximal_element fld amat acc =
- fld (fn (m,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
+ fld (fn (_,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x
in Real.fromInt a / Real.fromInt b end;
fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)
@@ -661,6 +703,7 @@
(* assignments for the others in terms of them. We give one pseudo-variable *)
(* "one" that's used for a constant term. *)
+(*
local
fun extract_first p l = case l of (* FIXME : use find_first instead *)
[] => error "extract_first"
@@ -686,10 +729,11 @@
fun tri_eliminate_equations one vars eqs =
let
val assig = eliminate vars Inttriplefunc.empty eqs
- val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ val vs = Inttriplefunc.fold (fn (_, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
in (distinct (dest_ord triple_int_ord) vs, assig)
end
end;
+*)
(* Eliminate all variables, in an essentially arbitrary order. *)
@@ -723,13 +767,14 @@
in fn eqs =>
let
val assig = eliminate Inttriplefunc.empty eqs
- val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ val vs = Inttriplefunc.fold (fn (_, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
in (distinct (dest_ord triple_int_ord) vs,assig)
end
end;
(* Solve equations by assigning arbitrary numbers. *)
+(*
fun tri_solve_equations one eqs =
let
val (vars,assigs) = tri_eliminate_all_equations one eqs
@@ -741,6 +786,7 @@
in if forall (fn e => tri_equation_eval ass e =/ rat_0) eqs
then Inttriplefunc.delete_safe one ass else raise Sanity
end;
+*)
(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
@@ -754,6 +800,7 @@
(* Usual operations on equation-parametrized poly. *)
+(*
fun tri_epoly_cmul c l =
if c =/ rat_0 then Inttriplefunc.empty else Inttriplefunc.map (K (tri_equation_cmul c)) l;;
@@ -762,9 +809,11 @@
val tri_epoly_add = Inttriplefunc.combine tri_equation_add Inttriplefunc.is_empty;
fun tri_epoly_sub p q = tri_epoly_add p (tri_epoly_neg q);;
+*)
(* Stuff for "equations" ((int*int)->num functions). *)
+(*
fun pi_equation_cmul c eq =
if c =/ rat_0 then Inttriplefunc.empty else Inttriplefunc.map (fn _ => fn d => c */ d) eq;
@@ -774,11 +823,13 @@
let fun value v = Inttriplefunc.apply assig v
in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
end;
+*)
(* Eliminate among linear equations: return unconstrained variables and *)
(* assignments for the others in terms of them. We give one pseudo-variable *)
(* "one" that's used for a constant term. *)
+(*
local
fun extract_first p l = case l of
[] => error "extract_first"
@@ -804,13 +855,15 @@
fun pi_eliminate_equations one vars eqs =
let
val assig = eliminate vars Inttriplefunc.empty eqs
- val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ val vs = Inttriplefunc.fold (fn (_, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
in (distinct (dest_ord triple_int_ord) vs, assig)
end
end;
+*)
(* Eliminate all variables, in an essentially arbitrary order. *)
+(*
fun pi_eliminate_all_equations one =
let
fun choose_variable eq =
@@ -841,13 +894,15 @@
in fn eqs =>
let
val assig = eliminate Inttriplefunc.empty eqs
- val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ val vs = Inttriplefunc.fold (fn (_, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
in (distinct (dest_ord triple_int_ord) vs,assig)
end
end;
+*)
(* Solve equations by assigning arbitrary numbers. *)
+(*
fun pi_solve_equations one eqs =
let
val (vars,assigs) = pi_eliminate_all_equations one eqs
@@ -859,9 +914,11 @@
in if forall (fn e => pi_equation_eval ass e =/ rat_0) eqs
then Inttriplefunc.delete_safe one ass else raise Sanity
end;
+*)
(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
+(*
fun pi_epoly_pmul p q acc =
FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>
FuncUtil.Monomialfunc.fold (fn (m2,e) => fn b =>
@@ -869,9 +926,11 @@
val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
in FuncUtil.Monomialfunc.update (m,pi_equation_add (pi_equation_cmul c e) es) b
end) q a) p acc ;
+*)
(* Usual operations on equation-parametrized poly. *)
+(*
fun pi_epoly_cmul c l =
if c =/ rat_0 then Inttriplefunc.empty else Inttriplefunc.map (K (pi_equation_cmul c)) l;;
@@ -882,6 +941,7 @@
fun pi_epoly_sub p q = pi_epoly_add p (pi_epoly_neg q);;
fun allpairs f l1 l2 = fold_rev (fn x => (curry (op @)) (map (f x) l2)) l1 [];
+*)
(* Hence produce the "relevant" monomials: those whose squares lie in the *)
(* Newton polytope of the monomials in the input. (This is enough according *)
@@ -930,17 +990,20 @@
end
end;
+(*
fun gcd_rat a b = Rat.rat_of_int (Integer.gcd (int_of_rat a) (int_of_rat b));
+*)
(* Adjust a diagonalization to collect rationals at the start. *)
(* FIXME : Potentially polymorphic keys, but here only: integers!! *)
+(*
local
fun upd0 x y a = if y =/ rat_0 then a else FuncUtil.Intfunc.update(x,y) a;
fun mapa f (d,v) =
(d, FuncUtil.Intfunc.fold (fn (i,c) => fn a => upd0 i (f c) a) v FuncUtil.Intfunc.empty)
fun adj (c,l) =
let val a =
- FuncUtil.Intfunc.fold (fn (i,c) => fn a => lcm_rat a (denominator_rat c))
+ FuncUtil.Intfunc.fold (fn (_,c) => fn a => lcm_rat a (denominator_rat c))
(snd l) rat_1 //
FuncUtil.Intfunc.fold (fn (i,c) => fn a => gcd_rat a (numerator_rat c))
(snd l) rat_0
@@ -954,6 +1017,7 @@
in ((rat_1 // a),map (fn (c,l) => (a */ c,l)) d')
end
end;
+*)
(* Enumeration of monomials with given multidegree bound. *)
@@ -1031,7 +1095,9 @@
else Inttriplefunc.map (fn _ => fn x => c */ x) bm;
val bmatrix_neg = bmatrix_cmul (Rat.rat_of_int ~1);
+(*
fun bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;
+*)
(* Smash a block matrix into components. *)
@@ -1039,14 +1105,14 @@
map (fn (bs,b0) =>
let val m = Inttriplefunc.fold
(fn ((b,i,j),c) => fn a => if b = b0 then FuncUtil.Intpairfunc.update ((i,j),c) a else a) bm FuncUtil.Intpairfunc.empty
- val d = FuncUtil.Intpairfunc.fold (fn ((i,j),c) => fn a => max a (max i j)) m 0
+ val _ = FuncUtil.Intpairfunc.fold (fn ((i,j),_) => fn a => max a (max i j)) m 0
in (((bs,bs),m):matrix) end)
(blocksizes ~~ (1 upto length blocksizes));;
(* FIXME : Get rid of this !!!*)
local
- fun tryfind_with msg f [] = raise Failure msg
- | tryfind_with msg f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
+ fun tryfind_with msg _ [] = raise Failure msg
+ | tryfind_with _ f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
in
fun tryfind f = tryfind_with "tryfind" f
end
@@ -1060,7 +1126,7 @@
(pol :: eqs @ map fst leqs) []
val monoid = if linf then
(poly_const rat_1,RealArith.Rational_lt rat_1)::
- (filter (fn (p,c) => multidegree p <= d) leqs)
+ (filter (fn (p,_) => multidegree p <= d) leqs)
else enumerate_products d leqs
val nblocks = length monoid
fun mk_idmultiplier k p =
@@ -1072,7 +1138,7 @@
fold_rev (fn (m,n) => FuncUtil.Monomialfunc.update(m,Inttriplefunc.onefunc((~k,~n,n),rat_1))) nons FuncUtil.Monomialfunc.empty)
end
- fun mk_sqmultiplier k (p,c) =
+ fun mk_sqmultiplier k (p,_) =
let
val e = (d - multidegree p) div 2
val mons = enumerate_monomials e vars
@@ -1091,13 +1157,13 @@
end
val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)
- val (idmonlist,ids) = split_list(map2 mk_idmultiplier (1 upto length eqs) eqs)
+ val (_(*idmonlist*),ids) = split_list(map2 mk_idmultiplier (1 upto length eqs) eqs)
val blocksizes = map length sqmonlist
val bigsum =
fold_rev2 (fn p => fn q => fn a => tri_epoly_pmul p q a) eqs ids
- (fold_rev2 (fn (p,c) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs
+ (fold_rev2 (fn (p,_) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs
(epoly_of_poly(poly_neg pol)))
- val eqns = FuncUtil.Monomialfunc.fold (fn (m,e) => fn a => e::a) bigsum []
+ val eqns = FuncUtil.Monomialfunc.fold (fn (_,e) => fn a => e::a) bigsum []
val (pvs,assig) = tri_eliminate_all_equations (0,0,0) eqns
val qvars = (0,0,0)::pvs
val allassig = fold_rev (fn v => Inttriplefunc.update(v,(Inttriplefunc.onefunc(v,rat_1)))) pvs assig
@@ -1119,7 +1185,7 @@
FuncUtil.Intfunc.empty)
val raw_vec = if null pvs then vector_0 0
else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats
- fun int_element (d,v) i = FuncUtil.Intfunc.tryapplyd v i rat_0
+ fun int_element (_,v) i = FuncUtil.Intfunc.tryapplyd v i rat_0
fun find_rounding d =
let
@@ -1154,10 +1220,10 @@
end
val sqs = map2 mk_sos sqmonlist ratdias
val cfs = map poly_of_epoly ids
- val msq = filter (fn (a,b) => not (null b)) (map2 pair monoid sqs)
+ val msq = filter (fn (_,b) => not (null b)) (map2 pair monoid sqs)
fun eval_sq sqs = fold_rev (fn (c,q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0
val sanity =
- fold_rev (fn ((p,c),s) => poly_add (poly_mul p (eval_sq s))) msq
+ fold_rev (fn ((p,_),s) => poly_add (poly_mul p (eval_sq s))) msq
(fold_rev2 (fn p => fn q => poly_add (poly_mul p q)) cfs eqs
(poly_neg pol))
@@ -1189,11 +1255,11 @@
(* FIXME: Replace tryfind by get_first !! *)
fun real_nonlinear_prover proof_method ctxt =
let
- val {add,mul,neg,pow,sub,main} = Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+ val {add = _, mul = _, neg = _, pow = _,
+ sub = _, main = real_poly_conv} =
+ Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
- val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
- real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
fun mainf cert_choice translator (eqs,les,lts) =
let
val eq0 = map (poly_of_term o Thm.dest_arg1 o concl) eqs
@@ -1244,9 +1310,9 @@
(poly_neg(poly_pow pol i))))
(0 upto k)
end
- val (d,i,(cert_ideal,cert_cone)) = deepen tryall 0
+ val (_,i,(cert_ideal,cert_cone)) = deepen tryall 0
val proofs_ideal =
- map2 (fn q => fn (p,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq
+ map2 (fn q => fn (_,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq
val proofs_cone = map cterm_of_sos cert_cone
val proof_ne = if null ltp then RealArith.Rational_lt Rat.one else
let val p = foldr1 RealArith.Product (map snd ltp)
@@ -1283,10 +1349,10 @@
fun substitutable_monomial fvs tm = case term_of tm of
Free(_,@{typ real}) => if not (member (op aconvc) fvs tm) then (Rat.one,tm)
else raise Failure "substitutable_monomial"
- | @{term "op * :: real => _"}$c$(t as Free _ ) =>
+ | @{term "op * :: real => _"}$_$(Free _) =>
if RealArith.is_ratconst (Thm.dest_arg1 tm) andalso not (member (op aconvc) fvs (Thm.dest_arg tm))
then (RealArith.dest_ratconst (Thm.dest_arg1 tm),Thm.dest_arg tm) else raise Failure "substitutable_monomial"
- | @{term "op + :: real => _"}$s$t =>
+ | @{term "op + :: real => _"}$_$_ =>
(substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg tm) fvs) (Thm.dest_arg1 tm)
handle Failure _ => substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg1 tm) fvs) (Thm.dest_arg tm))
| _ => raise Failure "substitutable_monomial"
@@ -1295,7 +1361,7 @@
let val w = Thm.dest_arg1 (cprop_of th)
in if v aconvc w then th
else case term_of w of
- @{term "op + :: real => _"}$s$t =>
+ @{term "op + :: real => _"}$_$_ =>
if Thm.dest_arg1 w aconvc v then shuffle2 th
else isolate_variable v (shuffle1 th)
| _ => error "isolate variable : This should not happen?"
@@ -1304,13 +1370,12 @@
fun real_nonlinear_subst_prover prover ctxt =
let
- val {add,mul,neg,pow,sub,main} = Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+ val {add = _, mul = real_poly_mul_conv, neg = _,
+ pow = _, sub = _, main = real_poly_conv} =
+ Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
- val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
- real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
-
fun make_substitution th =
let
val (c,v) = substitutable_monomial [] (Thm.dest_arg1(concl th))
@@ -1390,11 +1455,11 @@
val _ = print_cert certificates
in rtac ths 1 end)
-fun default_SOME f NONE v = SOME v
- | default_SOME f (SOME v) _ = SOME v;
+fun default_SOME _ NONE v = SOME v
+ | default_SOME _ (SOME v) _ = SOME v;
fun lift_SOME f NONE a = f a
- | lift_SOME f (SOME a) _ = SOME a;
+ | lift_SOME _ (SOME a) _ = SOME a;
local