(* Title: HOL/Library/positivstellensatz.ML
Author: Amine Chaieb, University of Cambridge
A generic arithmetic prover based on Positivstellensatz certificates
--- also implements Fourrier-Motzkin elimination as a special case
Fourrier-Motzkin elimination.
*)
(* A functor for finite mappings based on Tables *)
signature FUNC =
sig
include TABLE
val apply : 'a table -> key -> 'a
val applyd :'a table -> (key -> 'a) -> key -> 'a
val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
val dom : 'a table -> key list
val tryapplyd : 'a table -> key -> 'a -> 'a
val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
val choose : 'a table -> key * 'a
val onefunc : key * 'a -> 'a table
end;
functor FuncFun(Key: KEY) : FUNC=
struct
structure Tab = Table(Key);
open Tab;
fun dom a = sort Key.ord (Tab.keys a);
fun applyd f d x = case Tab.lookup f x of
SOME y => y
| NONE => d x;
fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
fun tryapplyd f a d = applyd f (K d) a;
fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
fun combine f z a b =
let
fun h (k,v) t = case Tab.lookup t k of
NONE => Tab.update (k,v) t
| SOME v' => let val w = f v v'
in if z w then Tab.delete k t else Tab.update (k,w) t end;
in Tab.fold h a b end;
fun choose f = case Tab.min_key f of
SOME k => (k, the (Tab.lookup f k))
| NONE => error "FuncFun.choose : Completely empty function"
fun onefunc kv = update kv empty
end;
(* Some standard functors and utility functions for them *)
structure FuncUtil =
struct
structure Intfunc = FuncFun(type key = int val ord = int_ord);
structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
val cterm_ord = Term_Ord.fast_term_ord o pairself term_of
structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
type monomial = int Ctermfunc.table;
val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
type poly = Rat.rat Monomialfunc.table;
(* The ordering so we can create canonical HOL polynomials. *)
fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
fun monomial_order (m1,m2) =
if Ctermfunc.is_empty m2 then LESS
else if Ctermfunc.is_empty m1 then GREATER
else
let val mon1 = dest_monomial m1
val mon2 = dest_monomial m2
val deg1 = fold (Integer.add o snd) mon1 0
val deg2 = fold (Integer.add o snd) mon2 0
in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
end;
end
(* positivstellensatz datatype and prover generation *)
signature REAL_ARITH =
sig
datatype positivstellensatz =
Axiom_eq of int
| Axiom_le of int
| Axiom_lt of int
| Rational_eq of Rat.rat
| Rational_le of Rat.rat
| Rational_lt of Rat.rat
| Square of FuncUtil.poly
| Eqmul of FuncUtil.poly * positivstellensatz
| Sum of positivstellensatz * positivstellensatz
| Product of positivstellensatz * positivstellensatz;
datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
datatype tree_choice = Left | Right
type prover = tree_choice list ->
(thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree
type cert_conv = cterm -> thm * pss_tree
val gen_gen_real_arith :
Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
conv * conv * conv * conv * conv * conv * prover -> cert_conv
val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree
val gen_real_arith : Proof.context ->
(Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
val gen_prover_real_arith : Proof.context -> prover -> cert_conv
val is_ratconst : cterm -> bool
val dest_ratconst : cterm -> Rat.rat
val cterm_of_rat : Rat.rat -> cterm
end
structure RealArith : REAL_ARITH =
struct
open Conv
(* ------------------------------------------------------------------------- *)
(* Data structure for Positivstellensatz refutations. *)
(* ------------------------------------------------------------------------- *)
datatype positivstellensatz =
Axiom_eq of int
| Axiom_le of int
| Axiom_lt of int
| Rational_eq of Rat.rat
| Rational_le of Rat.rat
| Rational_lt of Rat.rat
| Square of FuncUtil.poly
| Eqmul of FuncUtil.poly * positivstellensatz
| Sum of positivstellensatz * positivstellensatz
| Product of positivstellensatz * positivstellensatz;
(* Theorems used in the procedure *)
datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
datatype tree_choice = Left | Right
type prover = tree_choice list ->
(thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree
type cert_conv = cterm -> thm * pss_tree
(* Some useful derived rules *)
fun deduct_antisym_rule tha thb =
Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
(Thm.implies_intr (cprop_of tha) thb);
fun prove_hyp tha thb =
if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb) (* FIXME !? *)
then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
"((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
"((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
val pth_add =
@{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
"(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
"(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
"(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
"(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
val pth_mul =
@{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
"(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
"(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
"(x > 0 ==> y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
"(x > 0 ==> y > 0 ==> x * y > 0)"
by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
val pth_emul = @{lemma "y = (0::real) ==> x * y = 0" by simp};
val pth_square = @{lemma "x * x >= (0::real)" by simp};
val weak_dnf_simps =
List.take (simp_thms, 34) @
@{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
"(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
(*
val nnfD_simps =
@{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
"((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
"((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
*)
val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
val prenex_simps =
map (fn th => th RS sym)
([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
@{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
val real_abs_thms1 = @{lemma
"((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
"((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
"((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
"((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
"((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
"((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
"((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
"((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
"((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
"((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y + b >= r))" and
"((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
"((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y + c >= r))" and
"((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
"((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
"((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
"((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y + b >= r))" and
"((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
"((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y + c >= r))" and
"((min x y >= r) = (x >= r & y >= r))" and
"((min x y + a >= r) = (a + x >= r & a + y >= r))" and
"((a + min x y >= r) = (a + x >= r & a + y >= r))" and
"((a + min x y + b >= r) = (a + x + b >= r & a + y + b >= r))" and
"((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
"((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
"((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
"((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
"((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
"((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
"((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
"((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
"((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
"((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
"((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
"((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y + b > r))" and
"((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
"((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y + c > r))" and
"((min x y > r) = (x > r & y > r))" and
"((min x y + a > r) = (a + x > r & a + y > r))" and
"((a + min x y > r) = (a + x > r & a + y > r))" and
"((a + min x y + b > r) = (a + x + b > r & a + y + b > r))" and
"((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
"((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
by auto};
val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
by (atomize (full)) (auto split add: abs_split)};
val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
(* Miscellaneous *)
fun literals_conv bops uops cv =
let fun h t =
case (term_of t) of
b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
| u$_ => if member (op aconv) uops u then arg_conv h t else cv t
| _ => cv t
in h end;
fun cterm_of_rat x =
let val (a, b) = Rat.quotient_of_rat x
in
if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
(Numeral.mk_cnumber @{ctyp "real"} a))
(Numeral.mk_cnumber @{ctyp "real"} b)
end;
fun dest_ratconst t = case term_of t of
Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
| _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
fun is_ratconst t = can dest_ratconst t
(*
fun find_term p t = if p t then t else
case t of
a$b => (find_term p a handle TERM _ => find_term p b)
| Abs (_,_,t') => find_term p t'
| _ => raise TERM ("find_term",[t]);
*)
fun find_cterm p t = if p t then t else
case term_of t of
_$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
| Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
| _ => raise CTERM ("find_cterm",[t]);
(* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
fun is_comb t = case (term_of t) of _$_ => true | _ => false;
fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
handle CTERM _ => false;
(* Map back polynomials to HOL. *)
fun cterm_of_varpow x k = if k = 1 then x else Thm.capply (Thm.capply @{cterm "op ^ :: real => _"} x)
(Numeral.mk_cnumber @{ctyp nat} k)
fun cterm_of_monomial m =
if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
else
let
val m' = FuncUtil.dest_monomial m
val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
in foldr1 (fn (s, t) => Thm.capply (Thm.capply @{cterm "op * :: real => _"} s) t) vps
end
fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
else if c = Rat.one then cterm_of_monomial m
else Thm.capply (Thm.capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
fun cterm_of_poly p =
if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
else
let
val cms = map cterm_of_cmonomial
(sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
in foldr1 (fn (t1, t2) => Thm.capply(Thm.capply @{cterm "op + :: real => _"} t1) t2) cms
end;
(* A general real arithmetic prover *)
fun gen_gen_real_arith ctxt (mk_numeric,
numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
absconv1,absconv2,prover) =
let
val pre_ss = HOL_basic_ss addsimps simp_thms@ ex_simps@ all_simps@[@{thm not_all},@{thm not_ex},ex_disj_distrib, all_conj_distrib, @{thm if_bool_eq_disj}]
val prenex_ss = HOL_basic_ss addsimps prenex_simps
val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
fun oprconv cv ct =
let val g = Thm.dest_fun2 ct
in if g aconvc @{cterm "op <= :: real => _"}
orelse g aconvc @{cterm "op < :: real => _"}
then arg_conv cv ct else arg1_conv cv ct
end
fun real_ineq_conv th ct =
let
val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
end
val [real_lt_conv, real_le_conv, real_eq_conv,
real_not_lt_conv, real_not_le_conv, _] =
map real_ineq_conv pth
fun match_mp_rule ths ths' =
let
fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
| th::ths => (ths' MRS th handle THM _ => f ths ths')
in f ths ths' end
fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
(match_mp_rule pth_mul [th, th'])
fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
(match_mp_rule pth_add [th, th'])
fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
(instantiate' [] [SOME ct] (th RS pth_emul))
fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
(instantiate' [] [SOME t] pth_square)
fun hol_of_positivstellensatz(eqs,les,lts) proof =
let
fun translate prf = case prf of
Axiom_eq n => nth eqs n
| Axiom_le n => nth les n
| Axiom_lt n => nth lts n
| Rational_eq x => eqT_elim(numeric_eq_conv(Thm.capply @{cterm Trueprop}
(Thm.capply (Thm.capply @{cterm "op =::real => _"} (mk_numeric x))
@{cterm "0::real"})))
| Rational_le x => eqT_elim(numeric_ge_conv(Thm.capply @{cterm Trueprop}
(Thm.capply (Thm.capply @{cterm "op <=::real => _"}
@{cterm "0::real"}) (mk_numeric x))))
| Rational_lt x => eqT_elim(numeric_gt_conv(Thm.capply @{cterm Trueprop}
(Thm.capply (Thm.capply @{cterm "op <::real => _"} @{cterm "0::real"})
(mk_numeric x))))
| Square pt => square_rule (cterm_of_poly pt)
| Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
| Sum(p1,p2) => add_rule (translate p1) (translate p2)
| Product(p1,p2) => mul_rule (translate p1) (translate p2)
in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
(translate proof)
end
val init_conv = presimp_conv then_conv
nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
weak_dnf_conv
val concl = Thm.dest_arg o cprop_of
fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
val is_req = is_binop @{cterm "op =:: real => _"}
val is_ge = is_binop @{cterm "op <=:: real => _"}
val is_gt = is_binop @{cterm "op <:: real => _"}
val is_conj = is_binop @{cterm HOL.conj}
val is_disj = is_binop @{cterm HOL.disj}
fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
fun disj_cases th th1 th2 =
let val (p,q) = Thm.dest_binop (concl th)
val c = concl th1
val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
in Thm.implies_elim (Thm.implies_elim
(Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
(Thm.implies_intr (Thm.capply @{cterm Trueprop} p) th1))
(Thm.implies_intr (Thm.capply @{cterm Trueprop} q) th2)
end
fun overall cert_choice dun ths = case ths of
[] =>
let
val (eq,ne) = List.partition (is_req o concl) dun
val (le,nl) = List.partition (is_ge o concl) ne
val lt = filter (is_gt o concl) nl
in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
| th::oths =>
let
val ct = concl th
in
if is_conj ct then
let
val (th1,th2) = conj_pair th in
overall cert_choice dun (th1::th2::oths) end
else if is_disj ct then
let
val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
in (disj_cases th th1 th2, Branch (cert1, cert2)) end
else overall cert_choice (th::dun) oths
end
fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct
else raise CTERM ("dest_binary",[b,ct])
val dest_eq = dest_binary @{cterm "op = :: real => _"}
val neq_th = nth pth 5
fun real_not_eq_conv ct =
let
val (l,r) = dest_eq (Thm.dest_arg ct)
val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
val th' = Drule.binop_cong_rule @{cterm HOL.disj}
(Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
(Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
in Thm.transitive th th'
end
fun equal_implies_1_rule PQ =
let
val P = Thm.lhs_of PQ
in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
end
(* FIXME!!! Copied from groebner.ml *)
val strip_exists =
let fun h (acc, t) =
case (term_of t) of
Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t)
in fn t => h ([],t)
end
fun name_of x = case term_of x of
Free(s,_) => s
| Var ((s,_),_) => s
| _ => "x"
fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
fun choose v th th' = case concl_of th of
@{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
let
val p = (funpow 2 Thm.dest_arg o cprop_of) th
val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
val th0 = fconv_rule (Thm.beta_conversion true)
(instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
val pv = (Thm.rhs_of o Thm.beta_conversion true)
(Thm.capply @{cterm Trueprop} (Thm.capply p v))
val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
in Thm.implies_elim (Thm.implies_elim th0 th) th1 end
| _ => raise THM ("choose",0,[th, th'])
fun simple_choose v th =
choose v (Thm.assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
val strip_forall =
let fun h (acc, t) =
case (term_of t) of
Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t)
in fn t => h ([],t)
end
fun f ct =
let
val nnf_norm_conv' =
nnf_conv then_conv
literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
(Conv.cache_conv
(first_conv [real_lt_conv, real_le_conv,
real_eq_conv, real_not_lt_conv,
real_not_le_conv, real_not_eq_conv, all_conv]))
fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
(try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
val nct = Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} ct)
val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
val tm0 = Thm.dest_arg (Thm.rhs_of th0)
val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
let
val (evs,bod) = strip_exists tm0
val (avs,ibod) = strip_forall bod
val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.capply @{cterm Trueprop} bod))) th2)
in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
end
in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
end
in f
end;
(* A linear arithmetic prover *)
local
val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
val one_tm = @{cterm "1::real"}
fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
not(p(FuncUtil.Ctermfunc.apply e one_tm)))
fun linear_ineqs vars (les,lts) =
case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
SOME r => r
| NONE =>
(case find_first (contradictory (fn x => x >/ Rat.zero)) les of
SOME r => r
| NONE =>
if null vars then error "linear_ineqs: no contradiction" else
let
val ineqs = les @ lts
fun blowup v =
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
(map (fn v => (v,blowup v)) vars)))
fun addup (e1,p1) (e2,p2) acc =
let
val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
in if c1 */ c2 >=/ Rat.zero then acc else
let
val e1' = linear_cmul (Rat.abs c2) e1
val e2' = linear_cmul (Rat.abs c1) e2
val p1' = Product(Rational_lt(Rat.abs c2),p1)
val p2' = Product(Rational_lt(Rat.abs c1),p2)
in (linear_add e1' e2',Sum(p1',p2'))::acc
end
end
val (les0,les1) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
val (lts0,lts1) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
val (lesp,lesn) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
val (ltsp,ltsn) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
(fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
in linear_ineqs (remove (op aconvc) v vars) (les',lts')
end)
fun linear_eqs(eqs,les,lts) =
case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
SOME r => r
| NONE => (case eqs of
[] =>
let val vars = remove (op aconvc) one_tm
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
in linear_ineqs vars (les,lts) end
| (e,p)::es =>
if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
let
val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
fun xform (inp as (t,q)) =
let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
if d =/ Rat.zero then inp else
let
val k = (Rat.neg d) */ Rat.abs c // c
val e' = linear_cmul k e
val t' = linear_cmul (Rat.abs c) t
val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
val q' = Product(Rational_lt(Rat.abs c),q)
in (linear_add e' t',Sum(p',q'))
end
end
in linear_eqs(map xform es,map xform les,map xform lts)
end)
fun linear_prover (eq,le,lt) =
let
val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
val les = map_index (fn (n, p) => (p,Axiom_le n)) le
val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
in linear_eqs(eqs,les,lts)
end
fun lin_of_hol ct =
if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
else
let val (lop,r) = Thm.dest_comb ct
in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
else
let val (opr,l) = Thm.dest_comb lop
in if opr aconvc @{cterm "op + :: real =>_"}
then linear_add (lin_of_hol l) (lin_of_hol r)
else if opr aconvc @{cterm "op * :: real =>_"}
andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
end
end
fun is_alien ct = case term_of ct of
Const(@{const_name "real"}, _)$ n =>
if can HOLogic.dest_number n then false else true
| _ => false
in
fun real_linear_prover translator (eq,le,lt) =
let
val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
val eq_pols = map lhs eq
val le_pols = map rhs le
val lt_pols = map rhs lt
val aliens = filter is_alien
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
(eq_pols @ le_pols @ lt_pols) [])
val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
in ((translator (eq,le',lt) proof), Trivial)
end
end;
(* A less general generic arithmetic prover dealing with abs,max and min*)
local
val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
fun absmaxmin_elim_conv1 ctxt =
Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
val absmaxmin_elim_conv2 =
let
val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
val abs_tm = @{cterm "abs :: real => _"}
val p_tm = @{cpat "?P :: real => bool"}
val x_tm = @{cpat "?x :: real"}
val y_tm = @{cpat "?y::real"}
val is_max = is_binop @{cterm "max :: real => _"}
val is_min = is_binop @{cterm "min :: real => _"}
fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
fun eliminate_construct p c tm =
let
val t = find_cterm p tm
val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.capply (Thm.cabs t tm) t)
val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
(Thm.transitive th0 (c p ax))
end
val elim_abs = eliminate_construct is_abs
(fn p => fn ax =>
Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
val elim_max = eliminate_construct is_max
(fn p => fn ax =>
let val (ax,y) = Thm.dest_comb ax
in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
pth_max end)
val elim_min = eliminate_construct is_min
(fn p => fn ax =>
let val (ax,y) = Thm.dest_comb ax
in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
pth_min end)
in first_conv [elim_abs, elim_max, elim_min, all_conv]
end;
in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
end;
(* An instance for reals*)
fun gen_prover_real_arith ctxt prover =
let
fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
val {add, mul, neg, pow = _, sub = _, main} =
Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
in gen_real_arith ctxt
(cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
main,neg,add,mul, prover)
end;
end