order_tac: prevent potential bug, improve perf and tracing
- We need to be careful when the order operators contain schematic variables, e.g. <= = ccw' ?p.
structure Order_Procedure : sig
datatype inta = Int_of_integer of int
val integer_of_int : inta -> int
datatype 'a fm = Atom of 'a | And of 'a fm * 'a fm | Or of 'a fm * 'a fm |
Neg of 'a fm
datatype trm = Const of string | App of trm * trm | Var of inta
datatype prf_trm = PThm of string | Appt of prf_trm * trm |
AppP of prf_trm * prf_trm | AbsP of trm * prf_trm | Bound of trm |
Conv of trm * prf_trm * prf_trm
datatype order_atom = EQ of inta * inta | LEQ of inta * inta |
LESS of inta * inta
val lo_contr_prf : (bool * order_atom) fm -> prf_trm option
val po_contr_prf : (bool * order_atom) fm -> prf_trm option
end = struct
datatype inta = Int_of_integer of int;
fun integer_of_int (Int_of_integer k) = k;
fun equal_inta k l = integer_of_int k = integer_of_int l;
type 'a equal = {equal : 'a -> 'a -> bool};
val equal = #equal : 'a equal -> 'a -> 'a -> bool;
val equal_int = {equal = equal_inta} : inta equal;
fun less_eq_int k l = integer_of_int k <= integer_of_int l;
type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
val less_eq = #less_eq : 'a ord -> 'a -> 'a -> bool;
val less = #less : 'a ord -> 'a -> 'a -> bool;
fun less_int k l = integer_of_int k < integer_of_int l;
val ord_int = {less_eq = less_eq_int, less = less_int} : inta ord;
type 'a preorder = {ord_preorder : 'a ord};
val ord_preorder = #ord_preorder : 'a preorder -> 'a ord;
type 'a order = {preorder_order : 'a preorder};
val preorder_order = #preorder_order : 'a order -> 'a preorder;
val preorder_int = {ord_preorder = ord_int} : inta preorder;
val order_int = {preorder_order = preorder_int} : inta order;
type 'a linorder = {order_linorder : 'a order};
val order_linorder = #order_linorder : 'a linorder -> 'a order;
val linorder_int = {order_linorder = order_int} : inta linorder;
fun eq A_ a b = equal A_ a b;
fun equal_proda A_ B_ (x1, x2) (y1, y2) = eq A_ x1 y1 andalso eq B_ x2 y2;
fun equal_prod A_ B_ = {equal = equal_proda A_ B_} : ('a * 'b) equal;
fun less_eq_prod A_ B_ (x1, y1) (x2, y2) =
less A_ x1 x2 orelse less_eq A_ x1 x2 andalso less_eq B_ y1 y2;
fun less_prod A_ B_ (x1, y1) (x2, y2) =
less A_ x1 x2 orelse less_eq A_ x1 x2 andalso less B_ y1 y2;
fun ord_prod A_ B_ = {less_eq = less_eq_prod A_ B_, less = less_prod A_ B_} :
('a * 'b) ord;
fun preorder_prod A_ B_ =
{ord_preorder = ord_prod (ord_preorder A_) (ord_preorder B_)} :
('a * 'b) preorder;
fun order_prod A_ B_ =
{preorder_order = preorder_prod (preorder_order A_) (preorder_order B_)} :
('a * 'b) order;
fun linorder_prod A_ B_ =
{order_linorder = order_prod (order_linorder A_) (order_linorder B_)} :
('a * 'b) linorder;
datatype nat = Zero_nat | Suc of nat;
datatype color = R | B;
datatype ('a, 'b) rbta = Empty |
Branch of color * ('a, 'b) rbta * 'a * 'b * ('a, 'b) rbta;
datatype ('b, 'a) rbt = RBT of ('b, 'a) rbta;
datatype 'a set = Set of 'a list | Coset of 'a list;
datatype 'a fm = Atom of 'a | And of 'a fm * 'a fm | Or of 'a fm * 'a fm |
Neg of 'a fm;
datatype trm = Const of string | App of trm * trm | Var of inta;
datatype prf_trm = PThm of string | Appt of prf_trm * trm |
AppP of prf_trm * prf_trm | AbsP of trm * prf_trm | Bound of trm |
Conv of trm * prf_trm * prf_trm;
datatype ('a, 'b) mapping = Mapping of ('a, 'b) rbt;
datatype order_atom = EQ of inta * inta | LEQ of inta * inta |
LESS of inta * inta;
fun id x = (fn xa => xa) x;
fun impl_of B_ (RBT x) = x;
fun foldb f (Branch (c, lt, k, v, rt)) x = foldb f rt (f k v (foldb f lt x))
| foldb f Empty x = x;
fun fold A_ x xc = foldb x (impl_of A_ xc);
fun gen_keys kts (Branch (c, l, k, v, r)) = gen_keys ((k, r) :: kts) l
| gen_keys ((k, t) :: kts) Empty = k :: gen_keys kts t
| gen_keys [] Empty = [];
fun keysb x = gen_keys [] x;
fun keys A_ x = keysb (impl_of A_ x);
fun maps f [] = []
| maps f (x :: xs) = f x @ maps f xs;
fun empty A_ = RBT Empty;
fun foldl f a [] = a
| foldl f a (x :: xs) = foldl f (f a x) xs;
fun foldr f [] = id
| foldr f (x :: xs) = f x o foldr f xs;
fun balance (Branch (R, a, w, x, b)) s t (Branch (R, c, y, z, d)) =
Branch (R, Branch (B, a, w, x, b), s, t, Branch (B, c, y, z, d))
| balance (Branch (R, Branch (R, a, w, x, b), s, t, c)) y z Empty =
Branch (R, Branch (B, a, w, x, b), s, t, Branch (B, c, y, z, Empty))
| balance (Branch (R, Branch (R, a, w, x, b), s, t, c)) y z
(Branch (B, va, vb, vc, vd)) =
Branch
(R, Branch (B, a, w, x, b), s, t,
Branch (B, c, y, z, Branch (B, va, vb, vc, vd)))
| balance (Branch (R, Empty, w, x, Branch (R, b, s, t, c))) y z Empty =
Branch (R, Branch (B, Empty, w, x, b), s, t, Branch (B, c, y, z, Empty))
| balance
(Branch (R, Branch (B, va, vb, vc, vd), w, x, Branch (R, b, s, t, c))) y z
Empty =
Branch
(R, Branch (B, Branch (B, va, vb, vc, vd), w, x, b), s, t,
Branch (B, c, y, z, Empty))
| balance (Branch (R, Empty, w, x, Branch (R, b, s, t, c))) y z
(Branch (B, va, vb, vc, vd)) =
Branch
(R, Branch (B, Empty, w, x, b), s, t,
Branch (B, c, y, z, Branch (B, va, vb, vc, vd)))
| balance
(Branch (R, Branch (B, ve, vf, vg, vh), w, x, Branch (R, b, s, t, c))) y z
(Branch (B, va, vb, vc, vd)) =
Branch
(R, Branch (B, Branch (B, ve, vf, vg, vh), w, x, b), s, t,
Branch (B, c, y, z, Branch (B, va, vb, vc, vd)))
| balance Empty w x (Branch (R, b, s, t, Branch (R, c, y, z, d))) =
Branch (R, Branch (B, Empty, w, x, b), s, t, Branch (B, c, y, z, d))
| balance (Branch (B, va, vb, vc, vd)) w x
(Branch (R, b, s, t, Branch (R, c, y, z, d))) =
Branch
(R, Branch (B, Branch (B, va, vb, vc, vd), w, x, b), s, t,
Branch (B, c, y, z, d))
| balance Empty w x (Branch (R, Branch (R, b, s, t, c), y, z, Empty)) =
Branch (R, Branch (B, Empty, w, x, b), s, t, Branch (B, c, y, z, Empty))
| balance Empty w x
(Branch (R, Branch (R, b, s, t, c), y, z, Branch (B, va, vb, vc, vd))) =
Branch
(R, Branch (B, Empty, w, x, b), s, t,
Branch (B, c, y, z, Branch (B, va, vb, vc, vd)))
| balance (Branch (B, va, vb, vc, vd)) w x
(Branch (R, Branch (R, b, s, t, c), y, z, Empty)) =
Branch
(R, Branch (B, Branch (B, va, vb, vc, vd), w, x, b), s, t,
Branch (B, c, y, z, Empty))
| balance (Branch (B, va, vb, vc, vd)) w x
(Branch (R, Branch (R, b, s, t, c), y, z, Branch (B, ve, vf, vg, vh))) =
Branch
(R, Branch (B, Branch (B, va, vb, vc, vd), w, x, b), s, t,
Branch (B, c, y, z, Branch (B, ve, vf, vg, vh)))
| balance Empty s t Empty = Branch (B, Empty, s, t, Empty)
| balance Empty s t (Branch (B, va, vb, vc, vd)) =
Branch (B, Empty, s, t, Branch (B, va, vb, vc, vd))
| balance Empty s t (Branch (v, Empty, vb, vc, Empty)) =
Branch (B, Empty, s, t, Branch (v, Empty, vb, vc, Empty))
| balance Empty s t (Branch (v, Branch (B, ve, vf, vg, vh), vb, vc, Empty)) =
Branch
(B, Empty, s, t, Branch (v, Branch (B, ve, vf, vg, vh), vb, vc, Empty))
| balance Empty s t (Branch (v, Empty, vb, vc, Branch (B, vf, vg, vh, vi))) =
Branch
(B, Empty, s, t, Branch (v, Empty, vb, vc, Branch (B, vf, vg, vh, vi)))
| balance Empty s t
(Branch (v, Branch (B, ve, vj, vk, vl), vb, vc, Branch (B, vf, vg, vh, vi)))
= Branch
(B, Empty, s, t,
Branch
(v, Branch (B, ve, vj, vk, vl), vb, vc, Branch (B, vf, vg, vh, vi)))
| balance (Branch (B, va, vb, vc, vd)) s t Empty =
Branch (B, Branch (B, va, vb, vc, vd), s, t, Empty)
| balance (Branch (B, va, vb, vc, vd)) s t (Branch (B, ve, vf, vg, vh)) =
Branch (B, Branch (B, va, vb, vc, vd), s, t, Branch (B, ve, vf, vg, vh))
| balance (Branch (B, va, vb, vc, vd)) s t (Branch (v, Empty, vf, vg, Empty))
= Branch
(B, Branch (B, va, vb, vc, vd), s, t, Branch (v, Empty, vf, vg, Empty))
| balance (Branch (B, va, vb, vc, vd)) s t
(Branch (v, Branch (B, vi, vj, vk, vl), vf, vg, Empty)) =
Branch
(B, Branch (B, va, vb, vc, vd), s, t,
Branch (v, Branch (B, vi, vj, vk, vl), vf, vg, Empty))
| balance (Branch (B, va, vb, vc, vd)) s t
(Branch (v, Empty, vf, vg, Branch (B, vj, vk, vl, vm))) =
Branch
(B, Branch (B, va, vb, vc, vd), s, t,
Branch (v, Empty, vf, vg, Branch (B, vj, vk, vl, vm)))
| balance (Branch (B, va, vb, vc, vd)) s t
(Branch (v, Branch (B, vi, vn, vo, vp), vf, vg, Branch (B, vj, vk, vl, vm)))
= Branch
(B, Branch (B, va, vb, vc, vd), s, t,
Branch
(v, Branch (B, vi, vn, vo, vp), vf, vg, Branch (B, vj, vk, vl, vm)))
| balance (Branch (v, Empty, vb, vc, Empty)) s t Empty =
Branch (B, Branch (v, Empty, vb, vc, Empty), s, t, Empty)
| balance (Branch (v, Empty, vb, vc, Branch (B, ve, vf, vg, vh))) s t Empty =
Branch
(B, Branch (v, Empty, vb, vc, Branch (B, ve, vf, vg, vh)), s, t, Empty)
| balance (Branch (v, Branch (B, vf, vg, vh, vi), vb, vc, Empty)) s t Empty =
Branch
(B, Branch (v, Branch (B, vf, vg, vh, vi), vb, vc, Empty), s, t, Empty)
| balance
(Branch (v, Branch (B, vf, vg, vh, vi), vb, vc, Branch (B, ve, vj, vk, vl)))
s t Empty =
Branch
(B, Branch
(v, Branch (B, vf, vg, vh, vi), vb, vc, Branch (B, ve, vj, vk, vl)),
s, t, Empty)
| balance (Branch (v, Empty, vf, vg, Empty)) s t (Branch (B, va, vb, vc, vd))
= Branch
(B, Branch (v, Empty, vf, vg, Empty), s, t, Branch (B, va, vb, vc, vd))
| balance (Branch (v, Empty, vf, vg, Branch (B, vi, vj, vk, vl))) s t
(Branch (B, va, vb, vc, vd)) =
Branch
(B, Branch (v, Empty, vf, vg, Branch (B, vi, vj, vk, vl)), s, t,
Branch (B, va, vb, vc, vd))
| balance (Branch (v, Branch (B, vj, vk, vl, vm), vf, vg, Empty)) s t
(Branch (B, va, vb, vc, vd)) =
Branch
(B, Branch (v, Branch (B, vj, vk, vl, vm), vf, vg, Empty), s, t,
Branch (B, va, vb, vc, vd))
| balance
(Branch (v, Branch (B, vj, vk, vl, vm), vf, vg, Branch (B, vi, vn, vo, vp)))
s t (Branch (B, va, vb, vc, vd)) =
Branch
(B, Branch
(v, Branch (B, vj, vk, vl, vm), vf, vg, Branch (B, vi, vn, vo, vp)),
s, t, Branch (B, va, vb, vc, vd));
fun rbt_ins A_ f k v Empty = Branch (R, Empty, k, v, Empty)
| rbt_ins A_ f k v (Branch (B, l, x, y, r)) =
(if less A_ k x then balance (rbt_ins A_ f k v l) x y r
else (if less A_ x k then balance l x y (rbt_ins A_ f k v r)
else Branch (B, l, x, f k y v, r)))
| rbt_ins A_ f k v (Branch (R, l, x, y, r)) =
(if less A_ k x then Branch (R, rbt_ins A_ f k v l, x, y, r)
else (if less A_ x k then Branch (R, l, x, y, rbt_ins A_ f k v r)
else Branch (R, l, x, f k y v, r)));
fun paint c Empty = Empty
| paint c (Branch (uu, l, k, v, r)) = Branch (c, l, k, v, r);
fun rbt_insert_with_key A_ f k v t = paint B (rbt_ins A_ f k v t);
fun rbt_insert A_ = rbt_insert_with_key A_ (fn _ => fn _ => fn nv => nv);
fun insert A_ xc xd xe =
RBT (rbt_insert ((ord_preorder o preorder_order o order_linorder) A_) xc xd
(impl_of A_ xe));
fun rbt_lookup A_ Empty k = NONE
| rbt_lookup A_ (Branch (uu, l, x, y, r)) k =
(if less A_ k x then rbt_lookup A_ l k
else (if less A_ x k then rbt_lookup A_ r k else SOME y));
fun lookup A_ x =
rbt_lookup ((ord_preorder o preorder_order o order_linorder) A_)
(impl_of A_ x);
fun member A_ [] y = false
| member A_ (x :: xs) y = eq A_ x y orelse member A_ xs y;
fun remdups A_ [] = []
| remdups A_ (x :: xs) =
(if member A_ xs x then remdups A_ xs else x :: remdups A_ xs);
fun dnf_and_fm (Or (phi_1, phi_2)) psi =
Or (dnf_and_fm phi_1 psi, dnf_and_fm phi_2 psi)
| dnf_and_fm (Atom v) (Or (phi_1, phi_2)) =
Or (dnf_and_fm (Atom v) phi_1, dnf_and_fm (Atom v) phi_2)
| dnf_and_fm (And (v, va)) (Or (phi_1, phi_2)) =
Or (dnf_and_fm (And (v, va)) phi_1, dnf_and_fm (And (v, va)) phi_2)
| dnf_and_fm (Neg v) (Or (phi_1, phi_2)) =
Or (dnf_and_fm (Neg v) phi_1, dnf_and_fm (Neg v) phi_2)
| dnf_and_fm (Atom v) (Atom va) = And (Atom v, Atom va)
| dnf_and_fm (Atom v) (And (va, vb)) = And (Atom v, And (va, vb))
| dnf_and_fm (Atom v) (Neg va) = And (Atom v, Neg va)
| dnf_and_fm (And (v, va)) (Atom vb) = And (And (v, va), Atom vb)
| dnf_and_fm (And (v, va)) (And (vb, vc)) = And (And (v, va), And (vb, vc))
| dnf_and_fm (And (v, va)) (Neg vb) = And (And (v, va), Neg vb)
| dnf_and_fm (Neg v) (Atom va) = And (Neg v, Atom va)
| dnf_and_fm (Neg v) (And (va, vb)) = And (Neg v, And (va, vb))
| dnf_and_fm (Neg v) (Neg va) = And (Neg v, Neg va);
fun dnf_fm (And (phi_1, phi_2)) = dnf_and_fm (dnf_fm phi_1) (dnf_fm phi_2)
| dnf_fm (Or (phi_1, phi_2)) = Or (dnf_fm phi_1, dnf_fm phi_2)
| dnf_fm (Atom v) = Atom v
| dnf_fm (Neg v) = Neg v;
fun folda A_ f (Mapping t) a = fold A_ f t a;
fun keysa A_ (Mapping t) = Set (keys A_ t);
fun amap_fm h (Atom a) = h a
| amap_fm h (And (phi_1, phi_2)) = And (amap_fm h phi_1, amap_fm h phi_2)
| amap_fm h (Or (phi_1, phi_2)) = Or (amap_fm h phi_1, amap_fm h phi_2)
| amap_fm h (Neg phi) = Neg (amap_fm h phi);
fun emptya A_ = Mapping (empty A_);
fun lookupa A_ (Mapping t) = lookup A_ t;
fun update A_ k v (Mapping t) = Mapping (insert A_ k v t);
fun gen_length n (x :: xs) = gen_length (Suc n) xs
| gen_length n [] = n;
fun size_list x = gen_length Zero_nat x;
fun card A_ (Set xs) = size_list (remdups A_ xs);
fun conj_list (And (phi_1, phi_2)) = conj_list phi_1 @ conj_list phi_2
| conj_list (Atom a) = [a];
fun trm_of_fm f (Atom a) = f a
| trm_of_fm f (And (phi_1, phi_2)) =
App (App (Const "conj", trm_of_fm f phi_1), trm_of_fm f phi_2)
| trm_of_fm f (Or (phi_1, phi_2)) =
App (App (Const "disj", trm_of_fm f phi_1), trm_of_fm f phi_2)
| trm_of_fm f (Neg phi) = App (Const "Not", trm_of_fm f phi);
fun dnf_and_fm_prf (Or (phi_1, phi_2)) psi =
foldl (fn a => fn b => AppP (a, b)) (PThm "then_conv")
[PThm "conj_disj_distribR_conv",
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_and_fm_prf phi_1 psi),
dnf_and_fm_prf phi_2 psi]]
| dnf_and_fm_prf (Atom v) (Or (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "then_conv")
[PThm "conj_disj_distribL_conv",
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_and_fm_prf (Atom v) phi_1),
dnf_and_fm_prf (Atom v) phi_2]]
| dnf_and_fm_prf (And (v, va)) (Or (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "then_conv")
[PThm "conj_disj_distribL_conv",
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_and_fm_prf (And (v, va)) phi_1),
dnf_and_fm_prf (And (v, va)) phi_2]]
| dnf_and_fm_prf (Neg v) (Or (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "then_conv")
[PThm "conj_disj_distribL_conv",
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_and_fm_prf (Neg v) phi_1),
dnf_and_fm_prf (Neg v) phi_2]]
| dnf_and_fm_prf (Atom v) (Atom va) = PThm "all_conv"
| dnf_and_fm_prf (Atom v) (And (va, vb)) = PThm "all_conv"
| dnf_and_fm_prf (Atom v) (Neg va) = PThm "all_conv"
| dnf_and_fm_prf (And (v, va)) (Atom vb) = PThm "all_conv"
| dnf_and_fm_prf (And (v, va)) (And (vb, vc)) = PThm "all_conv"
| dnf_and_fm_prf (And (v, va)) (Neg vb) = PThm "all_conv"
| dnf_and_fm_prf (Neg v) (Atom va) = PThm "all_conv"
| dnf_and_fm_prf (Neg v) (And (va, vb)) = PThm "all_conv"
| dnf_and_fm_prf (Neg v) (Neg va) = PThm "all_conv";
fun dnf_fm_prf (And (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "then_conv")
[foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_fm_prf phi_1), dnf_fm_prf phi_2],
dnf_and_fm_prf (dnf_fm phi_1) (dnf_fm phi_2)]
| dnf_fm_prf (Or (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", dnf_fm_prf phi_1), dnf_fm_prf phi_2]
| dnf_fm_prf (Atom v) = PThm "all_conv"
| dnf_fm_prf (Neg v) = PThm "all_conv";
fun of_alist A_ xs = foldr (fn (a, b) => update A_ a b) xs (emptya A_);
fun deneg (true, LESS (x, y)) =
And (Atom (true, LEQ (x, y)), Atom (false, EQ (x, y)))
| deneg (false, LESS (x, y)) = Atom (true, LEQ (y, x))
| deneg (false, LEQ (x, y)) =
And (Atom (true, LEQ (y, x)), Atom (false, EQ (y, x)))
| deneg (false, EQ (v, vb)) = Atom (false, EQ (v, vb))
| deneg (v, EQ (vb, vc)) = Atom (v, EQ (vb, vc))
| deneg (true, LEQ (vb, vc)) = Atom (true, LEQ (vb, vc));
fun map_option f NONE = NONE
| map_option f (SOME x2) = SOME (f x2);
fun from_conj_prf trm_of_atom p (And (a, b)) =
foldl (fn aa => fn ba => AppP (aa, ba)) (PThm "conjE")
[Bound (trm_of_fm trm_of_atom (And (a, b))),
AbsP (trm_of_fm trm_of_atom a,
AbsP (trm_of_fm trm_of_atom b,
from_conj_prf trm_of_atom (from_conj_prf trm_of_atom p b)
a))]
| from_conj_prf trm_of_atom p (Atom a) = p;
fun contr_fm_prf trm_of_atom contr_atom_prf (Or (c, d)) =
(case contr_fm_prf trm_of_atom contr_atom_prf c of NONE => NONE
| SOME p1 =>
map_option
(fn p2 =>
foldl (fn a => fn b => AppP (a, b)) (PThm "disjE")
[Bound (trm_of_fm trm_of_atom (Or (c, d))),
AbsP (trm_of_fm trm_of_atom c, p1),
AbsP (trm_of_fm trm_of_atom d, p2)])
(contr_fm_prf trm_of_atom contr_atom_prf d))
| contr_fm_prf trm_of_atom contr_atom_prf (And (a, b)) =
(case contr_atom_prf (conj_list (And (a, b))) of NONE => NONE
| SOME p => SOME (from_conj_prf trm_of_atom p (And (a, b))))
| contr_fm_prf trm_of_atom contr_atom_prf (Atom a) = contr_atom_prf [a];
fun deless (true, LESS (x, y)) =
And (Atom (true, LEQ (x, y)), Atom (false, EQ (x, y)))
| deless (false, LESS (x, y)) =
Or (Atom (false, LEQ (x, y)), Atom (true, EQ (x, y)))
| deless (false, EQ (v, vb)) = Atom (false, EQ (v, vb))
| deless (false, LEQ (v, vb)) = Atom (false, LEQ (v, vb))
| deless (v, EQ (vb, vc)) = Atom (v, EQ (vb, vc))
| deless (v, LEQ (vb, vc)) = Atom (v, LEQ (vb, vc));
fun fst (x1, x2) = x1;
fun snd (x1, x2) = x2;
fun deneg_prf (true, LESS (x, y)) = PThm "less_le"
| deneg_prf (false, LESS (x, y)) = PThm "nless_le"
| deneg_prf (false, LEQ (x, y)) = PThm "nle_le"
| deneg_prf (false, EQ (v, vb)) = PThm "all_conv"
| deneg_prf (v, EQ (vb, vc)) = PThm "all_conv"
| deneg_prf (true, LEQ (vb, vc)) = PThm "all_conv";
val one_nat : nat = Suc Zero_nat;
fun deless_prf (true, LESS (x, y)) = PThm "less_le"
| deless_prf (false, LESS (x, y)) = PThm "nless_le"
| deless_prf (false, EQ (v, vb)) = PThm "all_conv"
| deless_prf (false, LEQ (v, vb)) = PThm "all_conv"
| deless_prf (v, EQ (vb, vc)) = PThm "all_conv"
| deless_prf (v, LEQ (vb, vc)) = PThm "all_conv";
fun minus_nat (Suc m) (Suc n) = minus_nat m n
| minus_nat Zero_nat n = Zero_nat
| minus_nat m Zero_nat = m;
fun relcomp1_mapping B_ (C1_, C2_) combine x y1 pxy pm pma =
folda (linorder_prod C2_ C2_)
(fn (y2, z) => fn pyz => fn pmb =>
(if eq C1_ y1 y2 andalso not (eq C1_ y2 z)
then update (linorder_prod B_ C2_) (x, z) (combine pxy pyz) pmb
else pmb))
pm pma;
fun relcomp_mapping (B1_, B2_) combine pm1 pm2 pma =
folda (linorder_prod B2_ B2_)
(fn (x, y) => fn pxy => fn pm =>
(if eq B1_ x y then pm
else relcomp1_mapping B2_ (B1_, B2_) combine x y pxy pm2 pm))
pm1 pma;
fun ntrancl_mapping (B1_, B2_) combine Zero_nat m = m
| ntrancl_mapping (B1_, B2_) combine (Suc k) m =
let
val trclm = ntrancl_mapping (B1_, B2_) combine k m;
in
relcomp_mapping (B1_, B2_) combine trclm m trclm
end;
fun trancl_mapping (B1_, B2_) combine m =
ntrancl_mapping (B1_, B2_) combine
(minus_nat (card (equal_prod B1_ B1_) (keysa (linorder_prod B2_ B2_) m))
one_nat)
m;
fun trm_of_order_atom (EQ (x, y)) = App (App (Const "eq", Var x), Var y)
| trm_of_order_atom (LEQ (x, y)) = App (App (Const "le", Var x), Var y)
| trm_of_order_atom (LESS (x, y)) = App (App (Const "lt", Var x), Var y);
fun trm_of_order_literal (true, a) = trm_of_order_atom a
| trm_of_order_literal (false, a) = App (Const "Not", trm_of_order_atom a);
fun is_in_leq leqm l =
(if equal_inta (fst l) (snd l) then SOME (Appt (PThm "refl", Var (fst l)))
else lookupa (linorder_prod linorder_int linorder_int) leqm l);
fun is_in_eq leqm l =
(case (is_in_leq leqm l, is_in_leq leqm (snd l, fst l)) of (NONE, _) => NONE
| (SOME _, NONE) => NONE
| (SOME p1, SOME p2) =>
SOME (foldl (fn a => fn b => AppP (a, b)) (PThm "antisym") [p1, p2]));
fun contr1_list leqm (false, LEQ (x, y)) =
map_option
(fn a =>
AppP (AppP (PThm "contr",
Bound (trm_of_order_literal (false, LEQ (x, y)))),
a))
(is_in_leq leqm (x, y))
| contr1_list leqm (false, EQ (x, y)) =
map_option
(fn a =>
AppP (AppP (PThm "contr",
Bound (trm_of_order_literal (false, EQ (x, y)))),
a))
(is_in_eq leqm (x, y))
| contr1_list uu (true, va) = NONE
| contr1_list uu (v, LESS (vb, vc)) = NONE;
fun contr_list_aux leqm [] = NONE
| contr_list_aux leqm (l :: ls) =
(case contr1_list leqm l of NONE => contr_list_aux leqm ls
| SOME a => SOME a);
fun leq1_member_list (true, LEQ (x, y)) =
[((x, y), Bound (trm_of_order_literal (true, LEQ (x, y))))]
| leq1_member_list (true, EQ (x, y)) =
[((x, y),
AppP (PThm "eqD1", Bound (trm_of_order_literal (true, EQ (x, y))))),
((y, x),
AppP (PThm "eqD2", Bound (trm_of_order_literal (true, EQ (x, y)))))]
| leq1_member_list (false, va) = []
| leq1_member_list (v, LESS (vb, vc)) = [];
fun leq1_list a = maps leq1_member_list a;
fun leq1_mapping a =
of_alist (linorder_prod linorder_int linorder_int) (leq1_list a);
fun contr_list a =
contr_list_aux
(trancl_mapping (equal_int, linorder_int)
(fn p1 => fn p2 =>
foldl (fn aa => fn b => AppP (aa, b)) (PThm "trans") [p1, p2])
(leq1_mapping a))
a;
fun contr_prf atom_conv phi =
contr_fm_prf trm_of_order_literal contr_list (dnf_fm (amap_fm atom_conv phi));
fun amap_f_m_prf ap (Atom a) = AppP (PThm "atom_conv", ap a)
| amap_f_m_prf ap (And (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", amap_f_m_prf ap phi_1), amap_f_m_prf ap phi_2]
| amap_f_m_prf ap (Or (phi_1, phi_2)) =
foldl (fn a => fn b => AppP (a, b)) (PThm "combination_conv")
[AppP (PThm "arg_conv", amap_f_m_prf ap phi_1), amap_f_m_prf ap phi_2]
| amap_f_m_prf ap (Neg phi) = AppP (PThm "arg_conv", amap_f_m_prf ap phi);
fun lo_contr_prf phi =
map_option
((fn a =>
Conv (trm_of_fm trm_of_order_literal phi, amap_f_m_prf deneg_prf phi,
a)) o
(fn a =>
Conv (trm_of_fm trm_of_order_literal (amap_fm deneg phi),
dnf_fm_prf (amap_fm deneg phi), a)))
(contr_prf deneg phi);
fun po_contr_prf phi =
map_option
((fn a =>
Conv (trm_of_fm trm_of_order_literal phi, amap_f_m_prf deless_prf phi,
a)) o
(fn a =>
Conv (trm_of_fm trm_of_order_literal (amap_fm deless phi),
dnf_fm_prf (amap_fm deless phi), a)))
(contr_prf deless phi);
end; (*struct Order_Procedure*)