src/Provers/linorder.ML
 changeset 14398 c5c47703f763 parent 14397 b3b15305a1c9 child 14399 dc677b35e54f
```     1.1 --- a/src/Provers/linorder.ML	Thu Feb 19 10:41:32 2004 +0100
1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,214 +0,0 @@
1.4 -(*
1.5 -  Title:	Transitivity reasoner for linear orders
1.6 -  Id:		\$Id\$
1.7 -  Author:	Clemens Ballarin, started 8 November 2002
1.9 -*)
1.10 -
1.11 -(***
1.12 -This is a very simple package for transitivity reasoning over linear orders.
1.13 -Simple means exponential time (and space) in the number of premises.
1.14 -Should be replaced by a graph-theoretic algorithm.
1.15 -
1.16 -The package provides a tactic trans_tac that uses all premises of the form
1.17 -
1.18 -  t = u, t < u, t <= u, ~(t < u) and ~(t <= u)
1.19 -
1.20 -to
1.21 -1. either derive a contradiction,
1.22 -   in which case the conclusion can be any term,
1.23 -2. or prove the conclusion, which must be of the same form as the premises.
1.24 -
1.25 -To get rid of ~= in the premises, it is advisable to use an elimination
1.26 -rule of the form
1.27 -
1.28 -  [| t ~= u; t < u ==> P; u < t ==> P |] ==> P.
1.29 -
1.30 -The package is implemented as an ML functor and thus not limited to the
1.31 -relation <= and friends.  It can be instantiated to any total order ---
1.32 -for example, the divisibility relation "dvd".
1.33 -***)
1.34 -
1.35 -(*** Credits ***
1.36 -
1.37 -This package is closely based on a (no longer used) transitivity reasoner for
1.38 -the natural numbers, which was written by Tobias Nipkow.
1.39 -
1.40 -****************)
1.41 -
1.42 -signature LESS_ARITH =
1.43 -sig
1.44 -  val less_reflE: thm  (* x < x ==> P *)
1.45 -  val le_refl: thm  (* x <= x *)
1.46 -  val less_imp_le: thm (* x < y ==> x <= y *)
1.47 -  val not_lessI: thm (* y <= x  ==> ~(x < y) *)
1.48 -  val not_leI: thm (* y < x  ==> ~(x <= y) *)
1.49 -  val not_lessD: thm (* ~(x < y) ==> y <= x *)
1.50 -  val not_leD: thm (* ~(x <= y) ==> y < x *)
1.51 -  val eqI: thm (* [| x <= y; y <= x |] ==> x = y *)
1.52 -  val eqD1: thm (* x = y ==> x <= y *)
1.53 -  val eqD2: thm (* x = y ==> y <= x *)
1.54 -  val less_trans: thm  (* [| x <= y; y <= z |] ==> x <= z *)
1.55 -  val less_le_trans: thm  (* [| x <= y; y < z |] ==> x < z *)
1.56 -  val le_less_trans: thm  (* [| x < y; y <= z |] ==> x < z *)
1.57 -  val le_trans: thm  (* [| x < y; y < z |] ==> x < z *)
1.58 -  val decomp: term -> (term * string * term) option
1.59 -    (* decomp (`x Rel y') should yield (x, Rel, y)
1.60 -       where Rel is one of "<", "<=", "~<", "~<=", "=" and "~="
1.61 -       other relation symbols are ignored *)
1.62 -end;
1.63 -
1.64 -signature TRANS_TAC =
1.65 -sig
1.66 -  val trans_tac: int -> tactic
1.67 -end;
1.68 -
1.69 -functor Trans_Tac_Fun (Less: LESS_ARITH): TRANS_TAC =
1.70 -struct
1.71 -
1.72 -(*** Proof objects ***)
1.73 -
1.74 -datatype proof
1.75 -  = Asm of int
1.76 -  | Thm of proof list * thm;
1.77 -
1.78 -(* Turn proof objects into theorems *)
1.79 -
1.80 -fun prove asms =
1.81 -  let fun pr (Asm i) = nth_elem (i, asms)
1.82 -        | pr (Thm (prfs, thm)) = (map pr prfs) MRS thm
1.83 -  in pr end;
1.84 -
1.85 -(*** Exceptions ***)
1.86 -
1.87 -exception Contr of proof;  (* Raised when contradiction is found *)
1.88 -
1.89 -exception Cannot;  (* Raised when goal cannot be proved *)
1.90 -
1.91 -(*** Internal representation of inequalities ***)
1.92 -
1.93 -datatype less
1.94 -  = Less of term * term * proof
1.95 -  | Le of term * term * proof;
1.96 -
1.97 -fun lower (Less (x, _, _)) = x
1.98 -  | lower (Le (x, _, _)) = x;
1.99 -
1.100 -fun upper (Less (_, y, _)) = y
1.101 -  | upper (Le (_, y, _)) = y;
1.102 -
1.103 -infix subsumes;
1.104 -
1.105 -fun (Less (x, y, _)) subsumes (Le (x', y', _)) = (x = x' andalso y = y')
1.106 -  | (Less (x, y, _)) subsumes (Less (x', y', _)) = (x = x' andalso y = y')
1.107 -  | (Le (x, y, _)) subsumes (Le (x', y', _)) = (x = x' andalso y = y')
1.108 -  | _ subsumes _ = false;
1.109 -
1.110 -fun trivial (Le (x, x', _)) = (x = x')
1.111 -  | trivial _ = false;
1.112 -
1.113 -(*** Transitive closure ***)
1.114 -
1.116 -  let fun adds([], news) = new::news
1.117 -        | adds(old::olds, news) = if new subsumes old then adds(olds, news)
1.120 -
1.121 -fun ctest (less as Less (x, x', p)) =
1.122 -    if x = x' then raise Contr (Thm ([p], Less.less_reflE))
1.123 -    else less
1.124 -  | ctest less = less;
1.125 -
1.126 -fun mktrans (Less (x, _, p), Less (_, z, q)) =
1.127 -    Less (x, z, Thm([p, q], Less.less_trans))
1.128 -  | mktrans (Less (x, _, p), Le (_, z, q)) =
1.129 -    Less (x, z, Thm([p, q], Less.less_le_trans))
1.130 -  | mktrans (Le (x, _, p), Less (_, z, q)) =
1.131 -    Less (x, z, Thm([p, q], Less.le_less_trans))
1.132 -  | mktrans (Le (x, _, p), Le (_, z, q)) =
1.133 -    Le (x, z, Thm([p, q], Less.le_trans));
1.134 -
1.135 -fun trans new olds =
1.136 -  let fun tr (news, old) =
1.137 -            if upper old = lower new then mktrans (old, new)::news
1.138 -            else if upper new = lower old then mktrans (new, old)::news
1.139 -            else news
1.140 -  in foldl tr ([], olds) end;
1.141 -
1.142 -fun close1 olds new =
1.143 -    if trivial new orelse exists (fn old => old subsumes new) olds then olds
1.144 -    else let val news = trans new olds
1.145 -         in close (add new (olds, [])) news end
1.146 -and close olds [] = olds
1.147 -  | close olds (new::news) = close (close1 olds (ctest new)) news;
1.148 -
1.149 -(*** Solving and proving goals ***)
1.150 -
1.151 -(* Recognise and solve trivial goal *)
1.152 -
1.153 -fun triv_sol (Le (x, x',  _)) =
1.154 -    if x = x' then Some (Thm ([], Less.le_refl))
1.155 -    else None
1.156 -  | triv_sol _ = None;
1.157 -
1.158 -(* Solve less starting from facts *)
1.159 -
1.160 -fun solve facts less =
1.161 -  case triv_sol less of
1.162 -    None => (case (Library.find_first (fn fact => fact subsumes less) facts, less) of
1.163 -	(None, _) => raise Cannot
1.164 -      | (Some (Less (_, _, p)), Less _) => p
1.165 -      | (Some (Le (_, _, p)), Less _) =>
1.166 -	   error "trans_tac/solve: internal error: le cannot subsume less"
1.167 -      | (Some (Less (_, _, p)), Le _) => Thm ([p], Less.less_imp_le)
1.168 -      | (Some (Le (_, _, p)), Le _) => p)
1.169 -  | Some prf => prf;
1.170 -
1.171 -(* Turn term t into Less or Le; n is position of t in list of assumptions *)
1.172 -
1.173 -fun mkasm (t, n) =
1.174 -  case Less.decomp t of
1.175 -    Some (x, rel, y) => (case rel of
1.176 -      "<"   => [Less (x, y, Asm n)]
1.177 -    | "~<"  => [Le (y, x, Thm ([Asm n], Less.not_lessD))]
1.178 -    | "<="  => [Le (x, y, Asm n)]
1.179 -    | "~<=" => [Less (y, x, Thm ([Asm n], Less.not_leD))]
1.180 -    | "="   => [Le (x, y, Thm ([Asm n], Less.eqD1)),
1.181 -                Le (x, y, Thm ([Asm n], Less.eqD1))]
1.182 -    | "~="  => []
1.183 -    | _     => error ("trans_tac/mkasm: unknown relation " ^ rel))
1.184 -  | None => [];
1.185 -
1.186 -(* Turn goal t into a pair (goals, proof) where goals is a list of
1.187 -   Le/Less-subgoals to solve, and proof the validation that proves the concl t
1.188 -   Asm ~1 is dummy (denotes a goal)
1.189 -*)
1.190 -
1.191 -fun mkconcl t =
1.192 -  case Less.decomp t of
1.193 -    Some (x, rel, y) => (case rel of
1.194 -      "<"   => ([Less (x, y, Asm ~1)], Asm 0)
1.195 -    | "~<"  => ([Le (y, x, Asm ~1)], Thm ([Asm 0], Less.not_lessI))
1.196 -    | "<="  => ([Le (x, y, Asm ~1)], Asm 0)
1.197 -    | "~<=" => ([Less (y, x, Asm ~1)], Thm ([Asm 0], Less.not_leI))
1.198 -    | "="   => ([Le (x, y, Asm ~1), Le (y, x, Asm ~1)],
1.199 -                 Thm ([Asm 0, Asm 1], Less.eqI))
1.200 -    | _  => raise Cannot)
1.201 -  | None => raise Cannot;
1.202 -
1.203 -val trans_tac = SUBGOAL (fn (A, n) =>
1.204 -  let val Hs = Logic.strip_assums_hyp A
1.205 -    val C = Logic.strip_assums_concl A
1.206 -    val lesss = flat (ListPair.map mkasm (Hs, 0 upto (length Hs - 1)))
1.207 -    val clesss = close [] lesss
1.208 -    val (subgoals, prf) = mkconcl C
1.209 -    val prfs = map (solve clesss) subgoals
1.210 -  in METAHYPS (fn asms =>
1.211 -    let val thms = map (prove asms) prfs
1.212 -    in rtac (prove thms prf) 1 end) n
1.213 -  end
1.214 -  handle Contr p => METAHYPS (fn asms => rtac (prove asms p) 1) n
1.215 -       | Cannot => no_tac);
1.216 -
1.217 -end;
```