--- a/src/HOL/Library/normarith.ML Tue May 12 17:32:49 2009 +0100
+++ b/src/HOL/Library/normarith.ML Tue May 12 17:32:49 2009 +0100
@@ -1,786 +1,7 @@
-(* A functor for finite mappings based on Tables *)
-signature FUNC =
-sig
- type 'a T
- type key
- val apply : 'a T -> key -> 'a
- val applyd :'a T -> (key -> 'a) -> key -> 'a
- val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
- val defined : 'a T -> key -> bool
- val dom : 'a T -> key list
- val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
- val graph : 'a T -> (key * 'a) list
- val is_undefined : 'a T -> bool
- val mapf : ('a -> 'b) -> 'a T -> 'b T
- val tryapplyd : 'a T -> key -> 'a -> 'a
- val undefine : key -> 'a T -> 'a T
- val undefined : 'a T
- val update : key * 'a -> 'a T -> 'a T
- val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
- val choose : 'a T -> key * 'a
- val onefunc : key * 'a -> 'a T
- val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
- val fns:
- {key_ord: key*key -> order,
- apply : 'a T -> key -> 'a,
- applyd :'a T -> (key -> 'a) -> key -> 'a,
- combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T,
- defined : 'a T -> key -> bool,
- dom : 'a T -> key list,
- fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b,
- graph : 'a T -> (key * 'a) list,
- is_undefined : 'a T -> bool,
- mapf : ('a -> 'b) -> 'a T -> 'b T,
- tryapplyd : 'a T -> key -> 'a -> 'a,
- undefine : key -> 'a T -> 'a T,
- undefined : 'a T,
- update : key * 'a -> 'a T -> 'a T,
- updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T,
- choose : 'a T -> key * 'a,
- onefunc : key * 'a -> 'a T,
- get_first: (key*'a -> 'a option) -> 'a T -> 'a option}
-end;
-
-functor FuncFun(Key: KEY) : FUNC=
-struct
-
-type key = Key.key;
-structure Tab = TableFun(Key);
-type 'a T = 'a Tab.table;
-
-val undefined = Tab.empty;
-val is_undefined = Tab.is_empty;
-val mapf = Tab.map;
-val fold = Tab.fold;
-val graph = Tab.dest;
-val dom = Tab.keys;
-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;
-val defined = Tab.defined;
-fun undefine x t = (Tab.delete x t handle UNDEF => t);
-val update = Tab.update;
-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.max_key f of
- SOME k => (k,valOf (Tab.lookup f k))
- | NONE => error "FuncFun.choose : Completely undefined function"
-
-fun onefunc kv = update kv undefined
-
-local
-fun find f (k,v) NONE = f (k,v)
- | find f (k,v) r = r
-in
-fun get_first f t = fold (find f) t NONE
-end
-
-val fns =
- {key_ord = Key.ord,
- apply = apply,
- applyd = applyd,
- combine = combine,
- defined = defined,
- dom = dom,
- fold = fold,
- graph = graph,
- is_undefined = is_undefined,
- mapf = mapf,
- tryapplyd = tryapplyd,
- undefine = undefine,
- undefined = undefined,
- update = update,
- updatep = updatep,
- choose = choose,
- onefunc = onefunc,
- get_first = get_first}
-
-end;
-
-structure Intfunc = FuncFun(type key = int val ord = int_ord);
-structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
-structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
-structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t)));
-structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
-
- (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
-structure Conv2 =
-struct
- open Conv
-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_abs t = case (term_of t) of Abs _ => true | _ => false;
-
-fun end_itlist f l =
- case l of
- [] => error "end_itlist"
- | [x] => x
- | (h::t) => f h (end_itlist f t);
-
- fun absc cv ct = case term_of ct of
- Abs (v,_, _) =>
- let val (x,t) = Thm.dest_abs (SOME v) ct
- in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
- end
- | _ => all_conv ct;
-
-fun cache_conv conv =
- let
- val tab = ref Termtab.empty
- fun cconv t =
- case Termtab.lookup (!tab) (term_of t) of
- SOME th => th
- | NONE => let val th = conv t
- in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
- in cconv end;
-fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
- handle CTERM _ => false;
-
-local
- fun thenqc conv1 conv2 tm =
- case try conv1 tm of
- SOME th1 => (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
- | NONE => conv2 tm
-
- fun thencqc conv1 conv2 tm =
- let val th1 = conv1 tm
- in (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
- end
- fun comb_qconv conv tm =
- let val (l,r) = Thm.dest_comb tm
- in (case try conv l of
- SOME th1 => (case try conv r of SOME th2 => Thm.combination th1 th2
- | NONE => Drule.fun_cong_rule th1 r)
- | NONE => Drule.arg_cong_rule l (conv r))
- end
- fun repeatqc conv tm = thencqc conv (repeatqc conv) tm
- fun sub_qconv conv tm = if is_abs tm then absc conv tm else comb_qconv conv tm
- fun once_depth_qconv conv tm =
- (conv else_conv (sub_qconv (once_depth_qconv conv))) tm
- fun depth_qconv conv tm =
- thenqc (sub_qconv (depth_qconv conv))
- (repeatqc conv) tm
- fun redepth_qconv conv tm =
- thenqc (sub_qconv (redepth_qconv conv))
- (thencqc conv (redepth_qconv conv)) tm
- fun top_depth_qconv conv tm =
- thenqc (repeatqc conv)
- (thencqc (sub_qconv (top_depth_qconv conv))
- (thencqc conv (top_depth_qconv conv))) tm
- fun top_sweep_qconv conv tm =
- thenqc (repeatqc conv)
- (sub_qconv (top_sweep_qconv conv)) tm
-in
-val (once_depth_conv, depth_conv, rdepth_conv, top_depth_conv, top_sweep_conv) =
- (fn c => try_conv (once_depth_qconv c),
- fn c => try_conv (depth_qconv c),
- fn c => try_conv (redepth_qconv c),
- fn c => try_conv (top_depth_qconv c),
- fn c => try_conv (top_sweep_qconv c));
-end;
-end;
-
-
- (* Some useful derived rules *)
-fun deduct_antisym_rule tha thb =
- equal_intr (implies_intr (cprop_of thb) tha)
- (implies_intr (cprop_of tha) thb);
-
-fun prove_hyp tha thb =
- if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb))
- then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
-
-
-
-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 cterm
- | Eqmul of cterm * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
-
-val gen_gen_real_arith :
- Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv *
- conv * conv * conv * conv * conv * conv *
- ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm) -> conv
-val real_linear_prover :
- (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm
-
-val gen_real_arith : Proof.context ->
- (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
- ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm) -> conv
-val gen_prover_real_arith : Proof.context ->
- ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
- thm list * thm list * thm list -> thm) -> conv
-val real_arith : Proof.context -> conv
-end
-
-structure RealArith (* : REAL_ARITH *)=
-struct
-
- open Conv Thm Conv2;;
-(* ------------------------------------------------------------------------- *)
-(* 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 cterm
- | Eqmul of cterm * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
- (* Theorems used in the procedure *)
-
-fun conjunctions th = case try Conjunction.elim th of
- SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
- | NONE => [th];
-
-val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0))
- &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
- &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
- by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |>
-conjunctions;
-
-val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
-val pth_add =
- @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0)
- &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0)
- &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0)
- &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0)
- &&& (x > 0 ==> y > 0 ==> x + y > 0)" by simp_all} |> conjunctions ;
-
-val pth_mul =
- @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&&
- (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&&
- (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
- (x > 0 ==> y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
- (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])} |> conjunctions;
-
-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)
- @ conjunctions @{lemma "((P & (Q | R)) = ((P&Q) | (P&R))) &&& ((Q | R) & P) = ((Q&P) | (R&P)) &&& (P & Q) = (Q & P) &&& ((P | Q) = (Q | P))" by blast+};
-
-val nnfD_simps = conjunctions @{lemma "((~(P & Q)) = (~P | ~Q)) &&& ((~(P | Q)) = (~P & ~Q) ) &&& ((P --> Q) = (~P | Q) ) &&& ((P = Q) = ((P & Q) | (~P & ~ Q))) &&& ((~(P = Q)) = ((P & ~ Q) | (~P & Q)) ) &&& ((~ ~(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 "all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
-
-val real_abs_thms1 = conjunctions @{lemma
- "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
- ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
- ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
- ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
- ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
- ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
- ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
- ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
- ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
- ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y + b >= r)) &&&
- ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
- ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y + c >= r)) &&&
- ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
- ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
- ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
- ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y + b >= r) )&&&
- ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
- ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y + c >= r)) &&&
- ((min x y >= r) = (x >= r & y >= r)) &&&
- ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
- ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
- ((a + min x y + b >= r) = (a + x + b >= r & a + y + b >= r)) &&&
- ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
- ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
- ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
- ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
- ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
- ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
- ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
- ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
- ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
- ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
- ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
- ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y + b > r)) &&&
- ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
- ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y + c > r)) &&&
- ((min x y > r) = (x > r & y > r)) &&&
- ((min x y + a > r) = (a + x > r & a + y > r)) &&&
- ((a + min x y > r) = (a + x > r & a + y > r)) &&&
- ((a + min x y + b > r) = (a + x + b > r & a + y + b > r)) &&&
- ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
- ((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::ordered_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)};
-
-
- (* Miscalineous *)
-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)
- | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
- | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
- fun is_ratconst t = can dest_ratconst t
-
-fun find_term p t = if p t then t else
- case t of
- a$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
- a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
- | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
- | _ => raise CTERM ("find_cterm",[t]);
-
-
- (* 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
- open Conv Thm;
- 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 = equal_elim (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' = (instantiate (match (lhs_of th, ct)) th
- handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
- in 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_mul_conv))
- (instantiate' [] [SOME t] pth_square)
-
- fun hol_of_positivstellensatz(eqs,les,lts) =
- 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(capply @{cterm Trueprop}
- (capply (capply @{cterm "op =::real => _"} (mk_numeric x))
- @{cterm "0::real"})))
- | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop}
- (capply (capply @{cterm "op <=::real => _"}
- @{cterm "0::real"}) (mk_numeric x))))
- | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop}
- (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
- (mk_numeric x))))
- | Square t => square_rule t
- | Eqmul(t,p) => emul_rule t (translate p)
- | Sum(p1,p2) => add_rule (translate p1) (translate p2)
- | Product(p1,p2) => mul_rule (translate p1) (translate p2)
- in fn prf =>
- fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
- (translate prf)
- 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 = dest_arg o cprop_of
- fun is_binop opr ct = (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 "op &"}
- val is_disj = is_binop @{cterm "op |"}
- fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
- fun disj_cases th th1 th2 =
- let val (p,q) = dest_binop (concl th)
- val c = concl th1
- val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
- in implies_elim (implies_elim (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) (implies_intr (capply @{cterm Trueprop} p) th1)) (implies_intr (capply @{cterm Trueprop} q) th2)
- end
- fun overall 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 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 dun (th1::th2::oths) end
- else if is_disj ct then
- let
- val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
- val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
- in disj_cases th th1 th2 end
- else overall (th::dun) oths
- end
- fun dest_binary b ct = if is_binop b ct then 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 (dest_arg ct)
- val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
- val th_p = poly_conv(dest_arg(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 "op |"}
- (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
- (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
- in transitive th th'
- end
- fun equal_implies_1_rule PQ =
- let
- val P = lhs_of PQ
- in implies_intr P (equal_elim PQ (assume P))
- end
- (* FIXME!!! Copied from groebner.ml *)
- val strip_exists =
- let fun h (acc, t) =
- case (term_of t) of
- Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (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) => _" }) (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("Ex",_)$_) =>
- let
- val p = (funpow 2 Thm.dest_arg o cprop_of) th
- val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
- val th0 = fconv_rule (Thm.beta_conversion true)
- (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
- val pv = (Thm.rhs_of o Thm.beta_conversion true)
- (Thm.capply @{cterm Trueprop} (Thm.capply p v))
- val th1 = forall_intr v (implies_intr pv th')
- in implies_elim (implies_elim th0 th) th1 end
- | _ => raise THM ("choose",0,[th, th'])
-
- fun simple_choose v th =
- choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
-
- val strip_forall =
- let fun h (acc, t) =
- case (term_of t) of
- Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (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 "op &"}, @{term "op |"}] []
- (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 "op &"}, @{term "op |"}] []
- (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 = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
- val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
- val tm0 = dest_arg (Thm.rhs_of th0)
- val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 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 = overall [] [specl avs (assume (Thm.rhs_of th1))]
- val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
- in Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
- end
- in implies_elim (instantiate' [] [SOME ct] pth_final) th
- end
-in f
-end;
-
-(* A linear arithmetic prover *)
-local
- val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
- fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
- val one_tm = @{cterm "1::real"}
- fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
- ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(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,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
- length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
- length(filter (fn (e,_) => 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 = Ctermfunc.tryapplyd e1 v Rat.zero
- val c2 = 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,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
- val (lts0,lts1) =
- List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
- val (lesp,lesn) =
- List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
- val (ltsp,ltsn) =
- List.partition (fn (e,_) => 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 (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) [])
- in linear_ineqs vars (les,lts) end
- | (e,p)::es =>
- if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
- let
- val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
- fun xform (inp as (t,q)) =
- let val d = 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(cterm_of_rat 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 = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
- val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
- val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
- in linear_eqs(eqs,les,lts)
- end
-
- fun lin_of_hol ct =
- if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
- else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
- else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
- else
- let val (lop,r) = Thm.dest_comb ct
- in if not (is_comb lop) then 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 Ctermfunc.onefunc (r, dest_ratconst l)
- else 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
- open Thm
-in
-fun real_linear_prover translator (eq,le,lt) =
- let
- val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
- val rhs = lin_of_hol o dest_arg o 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 (curry (gen_union (op aconvc)) o Ctermfunc.dom)
- (eq_pols @ le_pols @ lt_pols) [])
- val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
- val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
- val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
- in (translator (eq,le',lt) proof) : thm
- 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 dest_fun t aconvc abs_tm
- fun eliminate_construct p c tm =
- let
- val t = find_cterm p tm
- val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
- val (p,ax) = (dest_comb o Thm.rhs_of) th0
- in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
- (transitive th0 (c p ax))
- end
-
- val elim_abs = eliminate_construct is_abs
- (fn p => fn ax =>
- instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
- val elim_max = eliminate_construct is_max
- (fn p => fn ax =>
- let val (ax,y) = dest_comb ax
- in instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
- pth_max end)
- val elim_min = eliminate_construct is_min
- (fn p => fn ax =>
- let val (ax,y) = dest_comb ax
- in instantiate ([], [(p_tm,p), (x_tm, 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 = TermOrd.term_ord (term_of t, term_of u) = LESS
- val {add,mul,neg,pow,sub,main} =
- Normalizer.semiring_normalizers_ord_wrapper ctxt
- (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
- simple_cterm_ord
-in gen_real_arith ctxt
- (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
- main,neg,add,mul, prover)
-end;
-
-fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
-end
+(* Title: Library/normarith.ML
+ Author: Amine Chaieb, University of Cambridge
+ Description: A simple decision procedure for linear problems in euclidean space
+*)
(* Now the norm procedure for euclidean spaces *)