--- a/src/Provers/linorder.ML Thu Feb 19 10:41:32 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,214 +0,0 @@
-(*
- Title: Transitivity reasoner for linear orders
- Id: $Id$
- Author: Clemens Ballarin, started 8 November 2002
- Copyright: TU Muenchen
-*)
-
-(***
-This is a very simple package for transitivity reasoning over linear orders.
-Simple means exponential time (and space) in the number of premises.
-Should be replaced by a graph-theoretic algorithm.
-
-The package provides a tactic trans_tac that uses all premises of the form
-
- t = u, t < u, t <= u, ~(t < u) and ~(t <= u)
-
-to
-1. either derive a contradiction,
- in which case the conclusion can be any term,
-2. or prove the conclusion, which must be of the same form as the premises.
-
-To get rid of ~= in the premises, it is advisable to use an elimination
-rule of the form
-
- [| t ~= u; t < u ==> P; u < t ==> P |] ==> P.
-
-The package is implemented as an ML functor and thus not limited to the
-relation <= and friends. It can be instantiated to any total order ---
-for example, the divisibility relation "dvd".
-***)
-
-(*** Credits ***
-
-This package is closely based on a (no longer used) transitivity reasoner for
-the natural numbers, which was written by Tobias Nipkow.
-
-****************)
-
-signature LESS_ARITH =
-sig
- val less_reflE: thm (* x < x ==> P *)
- val le_refl: thm (* x <= x *)
- val less_imp_le: thm (* x < y ==> x <= y *)
- val not_lessI: thm (* y <= x ==> ~(x < y) *)
- val not_leI: thm (* y < x ==> ~(x <= y) *)
- val not_lessD: thm (* ~(x < y) ==> y <= x *)
- val not_leD: thm (* ~(x <= y) ==> y < x *)
- val eqI: thm (* [| x <= y; y <= x |] ==> x = y *)
- val eqD1: thm (* x = y ==> x <= y *)
- val eqD2: thm (* x = y ==> y <= x *)
- val less_trans: thm (* [| x <= y; y <= z |] ==> x <= z *)
- val less_le_trans: thm (* [| x <= y; y < z |] ==> x < z *)
- val le_less_trans: thm (* [| x < y; y <= z |] ==> x < z *)
- val le_trans: thm (* [| x < y; y < z |] ==> x < z *)
- val decomp: term -> (term * string * term) option
- (* decomp (`x Rel y') should yield (x, Rel, y)
- where Rel is one of "<", "<=", "~<", "~<=", "=" and "~="
- other relation symbols are ignored *)
-end;
-
-signature TRANS_TAC =
-sig
- val trans_tac: int -> tactic
-end;
-
-functor Trans_Tac_Fun (Less: LESS_ARITH): TRANS_TAC =
-struct
-
-(*** Proof objects ***)
-
-datatype proof
- = Asm of int
- | Thm of proof list * thm;
-
-(* Turn proof objects into theorems *)
-
-fun prove asms =
- let fun pr (Asm i) = nth_elem (i, asms)
- | pr (Thm (prfs, thm)) = (map pr prfs) MRS thm
- in pr end;
-
-(*** Exceptions ***)
-
-exception Contr of proof; (* Raised when contradiction is found *)
-
-exception Cannot; (* Raised when goal cannot be proved *)
-
-(*** Internal representation of inequalities ***)
-
-datatype less
- = Less of term * term * proof
- | Le of term * term * proof;
-
-fun lower (Less (x, _, _)) = x
- | lower (Le (x, _, _)) = x;
-
-fun upper (Less (_, y, _)) = y
- | upper (Le (_, y, _)) = y;
-
-infix subsumes;
-
-fun (Less (x, y, _)) subsumes (Le (x', y', _)) = (x = x' andalso y = y')
- | (Less (x, y, _)) subsumes (Less (x', y', _)) = (x = x' andalso y = y')
- | (Le (x, y, _)) subsumes (Le (x', y', _)) = (x = x' andalso y = y')
- | _ subsumes _ = false;
-
-fun trivial (Le (x, x', _)) = (x = x')
- | trivial _ = false;
-
-(*** Transitive closure ***)
-
-fun add new =
- let fun adds([], news) = new::news
- | adds(old::olds, news) = if new subsumes old then adds(olds, news)
- else adds(olds, old::news)
- in adds end;
-
-fun ctest (less as Less (x, x', p)) =
- if x = x' then raise Contr (Thm ([p], Less.less_reflE))
- else less
- | ctest less = less;
-
-fun mktrans (Less (x, _, p), Less (_, z, q)) =
- Less (x, z, Thm([p, q], Less.less_trans))
- | mktrans (Less (x, _, p), Le (_, z, q)) =
- Less (x, z, Thm([p, q], Less.less_le_trans))
- | mktrans (Le (x, _, p), Less (_, z, q)) =
- Less (x, z, Thm([p, q], Less.le_less_trans))
- | mktrans (Le (x, _, p), Le (_, z, q)) =
- Le (x, z, Thm([p, q], Less.le_trans));
-
-fun trans new olds =
- let fun tr (news, old) =
- if upper old = lower new then mktrans (old, new)::news
- else if upper new = lower old then mktrans (new, old)::news
- else news
- in foldl tr ([], olds) end;
-
-fun close1 olds new =
- if trivial new orelse exists (fn old => old subsumes new) olds then olds
- else let val news = trans new olds
- in close (add new (olds, [])) news end
-and close olds [] = olds
- | close olds (new::news) = close (close1 olds (ctest new)) news;
-
-(*** Solving and proving goals ***)
-
-(* Recognise and solve trivial goal *)
-
-fun triv_sol (Le (x, x', _)) =
- if x = x' then Some (Thm ([], Less.le_refl))
- else None
- | triv_sol _ = None;
-
-(* Solve less starting from facts *)
-
-fun solve facts less =
- case triv_sol less of
- None => (case (Library.find_first (fn fact => fact subsumes less) facts, less) of
- (None, _) => raise Cannot
- | (Some (Less (_, _, p)), Less _) => p
- | (Some (Le (_, _, p)), Less _) =>
- error "trans_tac/solve: internal error: le cannot subsume less"
- | (Some (Less (_, _, p)), Le _) => Thm ([p], Less.less_imp_le)
- | (Some (Le (_, _, p)), Le _) => p)
- | Some prf => prf;
-
-(* Turn term t into Less or Le; n is position of t in list of assumptions *)
-
-fun mkasm (t, n) =
- case Less.decomp t of
- Some (x, rel, y) => (case rel of
- "<" => [Less (x, y, Asm n)]
- | "~<" => [Le (y, x, Thm ([Asm n], Less.not_lessD))]
- | "<=" => [Le (x, y, Asm n)]
- | "~<=" => [Less (y, x, Thm ([Asm n], Less.not_leD))]
- | "=" => [Le (x, y, Thm ([Asm n], Less.eqD1)),
- Le (x, y, Thm ([Asm n], Less.eqD1))]
- | "~=" => []
- | _ => error ("trans_tac/mkasm: unknown relation " ^ rel))
- | None => [];
-
-(* Turn goal t into a pair (goals, proof) where goals is a list of
- Le/Less-subgoals to solve, and proof the validation that proves the concl t
- Asm ~1 is dummy (denotes a goal)
-*)
-
-fun mkconcl t =
- case Less.decomp t of
- Some (x, rel, y) => (case rel of
- "<" => ([Less (x, y, Asm ~1)], Asm 0)
- | "~<" => ([Le (y, x, Asm ~1)], Thm ([Asm 0], Less.not_lessI))
- | "<=" => ([Le (x, y, Asm ~1)], Asm 0)
- | "~<=" => ([Less (y, x, Asm ~1)], Thm ([Asm 0], Less.not_leI))
- | "=" => ([Le (x, y, Asm ~1), Le (y, x, Asm ~1)],
- Thm ([Asm 0, Asm 1], Less.eqI))
- | _ => raise Cannot)
- | None => raise Cannot;
-
-val trans_tac = SUBGOAL (fn (A, n) =>
- let val Hs = Logic.strip_assums_hyp A
- val C = Logic.strip_assums_concl A
- val lesss = flat (ListPair.map mkasm (Hs, 0 upto (length Hs - 1)))
- val clesss = close [] lesss
- val (subgoals, prf) = mkconcl C
- val prfs = map (solve clesss) subgoals
- in METAHYPS (fn asms =>
- let val thms = map (prove asms) prfs
- in rtac (prove thms prf) 1 end) n
- end
- handle Contr p => METAHYPS (fn asms => rtac (prove asms p) 1) n
- | Cannot => no_tac);
-
-end;