(* 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 (* Now the norm procedure for euclidean spaces *) signature NORM_ARITH = sig val norm_arith : Proof.context -> conv val norm_arith_tac : Proof.context -> int -> tactic end structure NormArith : NORM_ARITH = struct open Conv Thm Conv2; val bool_eq = op = : bool *bool -> bool 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 augment_norm b t acc = case term_of t of Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,dest_arg t) acc | _ => acc fun find_normedterms t acc = case term_of t of @{term "op + :: real => _"}$_$_ => find_normedterms (dest_arg1 t) (find_normedterms (dest_arg t) acc) | @{term "op * :: real => _"}$_$n => if not (is_ratconst (dest_arg1 t)) then acc else augment_norm (dest_ratconst (dest_arg1 t) >=/ Rat.zero) (dest_arg t) acc | _ => augment_norm true t acc val cterm_lincomb_neg = Ctermfunc.mapf Rat.neg fun cterm_lincomb_cmul c t = if c =/ Rat.zero then Ctermfunc.undefined else Ctermfunc.mapf (fn x => x */ c) t fun cterm_lincomb_add l r = Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r) fun cterm_lincomb_eq l r = Ctermfunc.is_undefined (cterm_lincomb_sub l r) val int_lincomb_neg = Intfunc.mapf Rat.neg fun int_lincomb_cmul c t = if c =/ Rat.zero then Intfunc.undefined else Intfunc.mapf (fn x => x */ c) t fun int_lincomb_add l r = Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r) fun int_lincomb_eq l r = Intfunc.is_undefined (int_lincomb_sub l r) fun vector_lincomb t = case term_of t of Const(@{const_name plus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ => cterm_lincomb_add (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t)) | Const(@{const_name minus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ => cterm_lincomb_sub (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t)) | Const(@{const_name vector_scalar_mult},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_$_ => cterm_lincomb_cmul (dest_ratconst (dest_arg1 t)) (vector_lincomb (dest_arg t)) | Const(@{const_name uminus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_ => cterm_lincomb_neg (vector_lincomb (dest_arg t)) | Const(@{const_name vec},_)$_ => let val b = ((snd o HOLogic.dest_number o term_of o dest_arg) t = 0 handle TERM _=> false) in if b then Ctermfunc.onefunc (t,Rat.one) else Ctermfunc.undefined end | _ => Ctermfunc.onefunc (t,Rat.one) fun vector_lincombs ts = fold_rev (fn t => fn fns => case AList.lookup (op aconvc) fns t of NONE => let val f = vector_lincomb t in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of SOME (_,f') => (t,f') :: fns | NONE => (t,f) :: fns end | SOME _ => fns) ts [] fun replacenegnorms cv t = case term_of t of @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t | @{term "op * :: real => _"}$_$_ => if dest_ratconst (dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t | _ => reflexive t fun flip v eq = if Ctermfunc.defined eq v then Ctermfunc.update (v, Rat.neg (Ctermfunc.apply eq v)) eq else eq fun allsubsets s = case s of [] => [[]] |(a::t) => let val res = allsubsets t in map (cons a) res @ res end fun evaluate env lin = Intfunc.fold (fn (x,c) => fn s => s +/ c */ (Intfunc.apply env x)) lin Rat.zero fun solve (vs,eqs) = case (vs,eqs) of ([],[]) => SOME (Intfunc.onefunc (0,Rat.one)) |(_,eq::oeqs) => (case filter (member (op =) vs) (Intfunc.dom eq) of (*FIXME use find_first here*) [] => NONE | v::_ => if Intfunc.defined eq v then let val c = Intfunc.apply eq v val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq fun eliminate eqn = if not (Intfunc.defined eqn v) then eqn else int_lincomb_add (int_lincomb_cmul (Intfunc.apply eqn v) vdef) eqn in (case solve (vs \ v,map eliminate oeqs) of NONE => NONE | SOME soln => SOME (Intfunc.update (v, evaluate soln (Intfunc.undefine v vdef)) soln)) end else NONE) fun combinations k l = if k = 0 then [[]] else case l of [] => [] | h::t => map (cons h) (combinations (k - 1) t) @ combinations k t fun forall2 p l1 l2 = case (l1,l2) of ([],[]) => true | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2 | _ => false; fun vertices vs eqs = let fun vertex cmb = case solve(vs,cmb) of NONE => NONE | SOME soln => SOME (map (fn v => Intfunc.tryapplyd soln v Rat.zero) vs) val rawvs = map_filter vertex (combinations (length vs) eqs) val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset [] end fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m fun subsume todo dun = case todo of [] => dun |v::ovs => let val dun' = if exists (fn w => subsumes w v) dun then dun else v::(filter (fn w => not(subsumes v w)) dun) in subsume ovs dun' end; fun match_mp PQ P = P RS PQ; 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 norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm}); fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm}; (* I think here the static context should be sufficient!! *) fun inequality_canon_rule ctxt = let (* FIXME : Should be computed statically!! *) val real_poly_conv = Normalizer.semiring_normalize_wrapper ctxt (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (field_comp_conv then_conv real_poly_conv))) end; 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 sub_conv cv ct = (comb_conv cv else_conv absc cv) ct; fun botc1 conv ct = ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct; fun rewrs_conv eqs ct = first_conv (map rewr_conv eqs) ct; val apply_pth1 = rewr_conv @{thm pth_1}; val apply_pth2 = rewr_conv @{thm pth_2}; val apply_pth3 = rewr_conv @{thm pth_3}; val apply_pth4 = rewrs_conv @{thms pth_4}; val apply_pth5 = rewr_conv @{thm pth_5}; val apply_pth6 = rewr_conv @{thm pth_6}; val apply_pth7 = rewrs_conv @{thms pth_7}; val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm vector_smult_lzero}))); val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv field_comp_conv); val apply_ptha = rewr_conv @{thm pth_a}; val apply_pthb = rewrs_conv @{thms pth_b}; val apply_pthc = rewrs_conv @{thms pth_c}; val apply_pthd = try_conv (rewr_conv @{thm pth_d}); fun headvector t = case t of Const(@{const_name plus}, Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$ (Const(@{const_name vector_scalar_mult}, _)$l$v)$r => v | Const(@{const_name vector_scalar_mult}, _)$l$v => v | _ => error "headvector: non-canonical term" fun vector_cmul_conv ct = ((apply_pth5 then_conv arg1_conv field_comp_conv) else_conv (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct fun vector_add_conv ct = apply_pth7 ct handle CTERM _ => (apply_pth8 ct handle CTERM _ => (case term_of ct of Const(@{const_name plus},_)$lt$rt => let val l = headvector lt val r = headvector rt in (case TermOrd.fast_term_ord (l,r) of LESS => (apply_pthb then_conv arg_conv vector_add_conv then_conv apply_pthd) ct | GREATER => (apply_pthc then_conv arg_conv vector_add_conv then_conv apply_pthd) ct | EQUAL => (apply_pth9 then_conv ((apply_ptha then_conv vector_add_conv) else_conv arg_conv vector_add_conv then_conv apply_pthd)) ct) end | _ => reflexive ct)) fun vector_canon_conv ct = case term_of ct of Const(@{const_name plus},_)$_$_ => let val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb val lth = vector_canon_conv l val rth = vector_canon_conv r val th = Drule.binop_cong_rule p lth rth in fconv_rule (arg_conv vector_add_conv) th end | Const(@{const_name vector_scalar_mult}, _)$_$_ => let val (p,r) = Thm.dest_comb ct val rth = Drule.arg_cong_rule p (vector_canon_conv r) in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth end | Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct | Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct | Const(@{const_name vec},_)$n => let val n = Thm.dest_arg ct in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero) then reflexive ct else apply_pth1 ct end | _ => apply_pth1 ct fun norm_canon_conv ct = case term_of ct of Const(@{const_name norm},_)$_ => arg_conv vector_canon_conv ct | _ => raise CTERM ("norm_canon_conv", [ct]) fun fold_rev2 f [] [] z = z | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z) | fold_rev2 f _ _ _ = raise UnequalLengths; fun int_flip v eq = if Intfunc.defined eq v then Intfunc.update (v, Rat.neg (Intfunc.apply eq v)) eq else eq; local val pth_zero = @{thm "norm_0"} val tv_n = (hd o tl o dest_ctyp o ctyp_of_term o dest_arg o dest_arg1 o dest_arg o cprop_of) pth_zero val concl = dest_arg o cprop_of fun real_vector_combo_prover ctxt translator (nubs,ges,gts) = let (* FIXME: Should be computed statically!!*) val real_poly_conv = Normalizer.semiring_normalize_wrapper ctxt (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) val sources = map (dest_arg o dest_arg1 o concl) nubs val rawdests = fold_rev (find_normedterms o dest_arg o concl) (ges @ gts) [] val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check" else () val dests = distinct (op aconvc) (map snd rawdests) val srcfuns = map vector_lincomb sources val destfuns = map vector_lincomb dests val vvs = fold_rev (curry (gen_union op aconvc) o Ctermfunc.dom) (srcfuns @ destfuns) [] val n = length srcfuns val nvs = 1 upto n val srccombs = srcfuns ~~ nvs fun consider d = let fun coefficients x = let val inp = if Ctermfunc.defined d x then Intfunc.onefunc (0, Rat.neg(Ctermfunc.apply d x)) else Intfunc.undefined in fold_rev (fn (f,v) => fn g => if Ctermfunc.defined f x then Intfunc.update (v, Ctermfunc.apply f x) g else g) srccombs inp end val equations = map coefficients vvs val inequalities = map (fn n => Intfunc.onefunc (n,Rat.one)) nvs fun plausiblevertices f = let val flippedequations = map (fold_rev int_flip f) equations val constraints = flippedequations @ inequalities val rawverts = vertices nvs constraints fun check_solution v = let val f = fold_rev2 (curry Intfunc.update) nvs v (Intfunc.onefunc (0, Rat.one)) in forall (fn e => evaluate f e =/ Rat.zero) flippedequations end val goodverts = filter check_solution rawverts val signfixups = map (fn n => if n mem_int f then ~1 else 1) nvs in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts end val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) [] in subsume allverts [] end fun compute_ineq v = let val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE else SOME(norm_cmul_rule v t)) (v ~~ nubs) in inequality_canon_rule ctxt (end_itlist norm_add_rule ths) end val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @ map (inequality_canon_rule ctxt) nubs @ ges val zerodests = filter (fn t => null (Ctermfunc.dom (vector_lincomb t))) (map snd rawdests) in RealArith.real_linear_prover translator (map (fn t => instantiate ([(tv_n,(hd o tl o dest_ctyp o ctyp_of_term) t)],[]) pth_zero) zerodests, map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv arg_conv (arg_conv real_poly_conv))) ges', map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv arg_conv (arg_conv real_poly_conv))) gts) end in val real_vector_combo_prover = real_vector_combo_prover end; local val pth = @{thm norm_imp_pos_and_ge} val norm_mp = match_mp pth val concl = dest_arg o cprop_of fun conjunct1 th = th RS @{thm conjunct1} fun conjunct2 th = th RS @{thm conjunct2} fun C f x y = f y x fun real_vector_ineq_prover ctxt translator (ges,gts) = let (* val _ = error "real_vector_ineq_prover: pause" *) val ntms = fold_rev find_normedterms (map (dest_arg o concl) (ges @ gts)) [] val lctab = vector_lincombs (map snd (filter (not o fst) ntms)) val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt fun mk_norm t = capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: norm) => real"}) t fun mk_equals l r = capply (capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r val asl = map2 (fn (t,_) => fn n => assume (mk_equals (mk_norm t) (cterm_of (ProofContext.theory_of ctxt') (Free(n,@{typ real}))))) lctab fxns val replace_conv = try_conv (rewrs_conv asl) val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv)) val ges' = fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths) asl (map replace_rule ges) val gts' = map replace_rule gts val nubs = map (conjunct2 o norm_mp) asl val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts') val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1)) val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1]) val cps = map (swap o dest_equals) (cprems_of th11) val th12 = instantiate ([], cps) th11 val th13 = fold (C implies_elim) (map (reflexive o snd) cps) th12; in hd (Variable.export ctxt' ctxt [th13]) end in val real_vector_ineq_prover = real_vector_ineq_prover end; local val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0})) fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS; (* FIXME: Lookup in the context every time!!! Fix this !!!*) fun splitequation ctxt th acc = let val real_poly_neg_conv = #neg (Normalizer.semiring_normalizers_ord_wrapper ctxt (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord) val (th1,th2) = conj_pair(rawrule th) in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc end in fun real_vector_prover ctxt translator (eqs,ges,gts) = real_vector_ineq_prover ctxt translator (fold_rev (splitequation ctxt) eqs ges,gts) end; fun init_conv ctxt = Simplifier.rewrite (Simplifier.context ctxt (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm dist_def}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths}))) then_conv field_comp_conv then_conv nnf_conv fun pure ctxt = RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt); fun norm_arith ctxt ct = let val ctxt' = Variable.declare_term (term_of ct) ctxt val th = init_conv ctxt' ct in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th)) (pure ctxt' (rhs_of th)) end fun norm_arith_tac ctxt = clarify_tac HOL_cs THEN' ObjectLogic.full_atomize_tac THEN' CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i); end;