(* Title: HOL/Tools/function_package/context_tree.ML
ID: $Id$
Author: Alexander Krauss, TU Muenchen
A package for general recursive function definitions.
Builds and traverses trees of nested contexts along a term.
*)
signature FUNDEF_CTXTREE =
sig
type ctx_tree
(* FIXME: This interface is a mess and needs to be cleaned up! *)
val cong_deps : thm -> int IntGraph.T
val add_congs : thm list
val mk_tree: (thm * FundefCommon.depgraph) list ->
(string * typ) -> Proof.context -> term -> FundefCommon.ctx_tree
val inst_tree: theory -> term -> term -> FundefCommon.ctx_tree
-> FundefCommon.ctx_tree
val add_context_varnames : FundefCommon.ctx_tree -> string list -> string list
val export_term : (string * typ) list * term list -> term -> term
val export_thm : theory -> (string * typ) list * term list -> thm -> thm
val import_thm : theory -> (string * typ) list * thm list -> thm -> thm
val traverse_tree :
((string * typ) list * thm list -> term ->
(((string * typ) list * thm list) * thm) list ->
(((string * typ) list * thm list) * thm) list * 'b ->
(((string * typ) list * thm list) * thm) list * 'b)
-> FundefCommon.ctx_tree -> 'b -> 'b
val rewrite_by_tree : theory -> term -> thm -> (thm * thm) list -> FundefCommon.ctx_tree -> thm * (thm * thm) list
end
structure FundefCtxTree : FUNDEF_CTXTREE =
struct
open FundefCommon
(* Maps "Trueprop A = B" to "A" *)
val rhs_of = snd o HOLogic.dest_eq o HOLogic.dest_Trueprop
(* Maps "A == B" to "B" *)
val meta_rhs_of = snd o Logic.dest_equals
(*** Dependency analysis for congruence rules ***)
fun branch_vars t =
let val (assumes, term) = dest_implies_list (snd (dest_all_all t))
in (fold (curry add_term_vars) assumes [], term_vars term)
end
fun cong_deps crule =
let
val branches = map branch_vars (prems_of crule)
val num_branches = (1 upto (length branches)) ~~ branches
in
IntGraph.empty
|> fold (fn (i,_)=> IntGraph.new_node (i,i)) num_branches
|> fold (fn ((i,(c1,_)),(j,(_, t2))) => if i = j orelse null (c1 inter t2) then I else IntGraph.add_edge_acyclic (i,j))
(product num_branches num_branches)
end
val add_congs = map (fn c => c RS eq_reflection) [cong, ext]
(* Called on the INSTANTIATED branches of the congruence rule *)
fun mk_branch ctx t =
let
val (ctx', fixes, impl) = dest_all_all_ctx ctx t
val (assumes, term) = dest_implies_list impl
in
(ctx', fixes, assumes, rhs_of term)
end
fun find_cong_rule ctx ((r,dep)::rs) t =
(let
val (c, subs) = (meta_rhs_of (concl_of r), prems_of r)
val subst = Pattern.match (ProofContext.theory_of ctx) (c, t) (Vartab.empty, Vartab.empty)
val branches = map (mk_branch ctx o Envir.beta_norm o Envir.subst_vars subst) subs
in
(r, dep, branches)
end
handle Pattern.MATCH => find_cong_rule ctx rs t)
| find_cong_rule _ [] _ = sys_error "function_package/context_tree.ML: No cong rule found!"
fun matchcall fvar (a $ b) = if a = Free fvar then SOME b else NONE
| matchcall fvar _ = NONE
fun mk_tree congs fvar ctx t =
case matchcall fvar t of
SOME arg => RCall (t, mk_tree congs fvar ctx arg)
| NONE =>
if not (exists_subterm (fn Free v => v = fvar | _ => false) t) then Leaf t
else
let val (r, dep, branches) = find_cong_rule ctx congs t in
Cong (t, r, dep,
map (fn (ctx', fixes, assumes, st) =>
(fixes, map (assume o cterm_of (ProofContext.theory_of ctx)) assumes,
mk_tree congs fvar ctx' st)) branches)
end
fun inst_tree thy fvar f tr =
let
val cfvar = cterm_of thy fvar
val cf = cterm_of thy f
fun inst_term t =
subst_bound(f, abstract_over (fvar, t))
val inst_thm = forall_elim cf o forall_intr cfvar
fun inst_tree_aux (Leaf t) = Leaf t
| inst_tree_aux (Cong (t, crule, deps, branches)) =
Cong (inst_term t, inst_thm crule, deps, map inst_branch branches)
| inst_tree_aux (RCall (t, str)) =
RCall (inst_term t, inst_tree_aux str)
and inst_branch (fxs, assms, str) =
(fxs, map (assume o cterm_of thy o inst_term o prop_of) assms, inst_tree_aux str)
in
inst_tree_aux tr
end
(* FIXME: remove *)
fun add_context_varnames (Leaf _) = I
| add_context_varnames (Cong (_, _, _, sub)) = fold (fn (fs, _, st) => fold (insert (op =) o fst) fs o add_context_varnames st) sub
| add_context_varnames (RCall (_,st)) = add_context_varnames st
(* Poor man's contexts: Only fixes and assumes *)
fun compose (fs1, as1) (fs2, as2) = (fs1 @ fs2, as1 @ as2)
fun export_term (fixes, assumes) =
fold_rev (curry Logic.mk_implies) assumes #> fold_rev (mk_forall o Free) fixes
fun export_thm thy (fixes, assumes) =
fold_rev (implies_intr o cterm_of thy) assumes
#> fold_rev (forall_intr o cterm_of thy o Free) fixes
fun import_thm thy (fixes, athms) =
fold (forall_elim o cterm_of thy o Free) fixes
#> fold implies_elim_swp athms
fun assume_in_ctxt thy (fixes, athms) prop =
let
val global_assum = export_term (fixes, map prop_of athms) prop
in
(global_assum,
assume (cterm_of thy global_assum) |> import_thm thy (fixes, athms))
end
(* folds in the order of the dependencies of a graph. *)
fun fold_deps G f x =
let
fun fill_table i (T, x) =
case Inttab.lookup T i of
SOME _ => (T, x)
| NONE =>
let
val (T', x') = fold fill_table (IntGraph.imm_succs G i) (T, x)
val (v, x'') = f (the o Inttab.lookup T') i x
in
(Inttab.update (i, v) T', x'')
end
val (T, x) = fold fill_table (IntGraph.keys G) (Inttab.empty, x)
in
(Inttab.fold (cons o snd) T [], x)
end
fun flatten xss = fold_rev append xss []
fun traverse_tree rcOp tr x =
let
fun traverse_help ctx (Leaf _) u x = ([], x)
| traverse_help ctx (RCall (t, st)) u x =
rcOp ctx t u (traverse_help ctx st u x)
| traverse_help ctx (Cong (t, crule, deps, branches)) u x =
let
fun sub_step lu i x =
let
val (fixes, assumes, subtree) = nth branches (i - 1)
val used = fold_rev (append o lu) (IntGraph.imm_succs deps i) u
val (subs, x') = traverse_help (compose ctx (fixes, assumes)) subtree used x
val exported_subs = map (apfst (compose (fixes, assumes))) subs
in
(exported_subs, x')
end
in
fold_deps deps sub_step x
|> apfst flatten
end
in
snd (traverse_help ([], []) tr [] x)
end
fun is_refl thm = let val (l,r) = Logic.dest_equals (prop_of thm) in l = r end
fun rewrite_by_tree thy h ih x tr =
let
fun rewrite_help fix f_as h_as x (Leaf t) = (reflexive (cterm_of thy t), x)
| rewrite_help fix f_as h_as x (RCall (_ $ arg, st)) =
let
val (inner, (lRi,ha)::x') = rewrite_help fix f_as h_as x st
(* Need not use the simplifier here. Can use primitive steps! *)
val rew_ha = if is_refl inner then I else simplify (HOL_basic_ss addsimps [inner])
val h_a_eq_h_a' = combination (reflexive (cterm_of thy h)) inner
val iha = import_thm thy (fix, h_as) ha (* (a', h a') : G *)
|> rew_ha
val inst_ih = instantiate' [] [SOME (cterm_of thy arg)] ih
val eq = implies_elim (implies_elim inst_ih lRi) iha
val h_a'_eq_f_a' = eq RS eq_reflection
val result = transitive h_a_eq_h_a' h_a'_eq_f_a'
in
(result, x')
end
| rewrite_help fix f_as h_as x (Cong (t, crule, deps, branches)) =
let
fun sub_step lu i x =
let
val (fixes, assumes, st) = nth branches (i - 1)
val used = fold_rev (cons o lu) (IntGraph.imm_succs deps i) []
val used_rev = map (fn u_eq => (u_eq RS sym) RS eq_reflection) used
val assumes' = map (simplify (HOL_basic_ss addsimps (filter_out is_refl used_rev))) assumes
val (subeq, x') = rewrite_help (fix @ fixes) (f_as @ assumes) (h_as @ assumes') x st
val subeq_exp = export_thm thy (fixes, map prop_of assumes) (subeq RS meta_eq_to_obj_eq)
in
(subeq_exp, x')
end
val (subthms, x') = fold_deps deps sub_step x
in
(fold_rev (curry op COMP) subthms crule, x')
end
in
rewrite_help [] [] [] x tr
end
end