(* Title: HOL/Import/shuffler.ML
Author: Sebastian Skalberg, TU Muenchen
Package for proving two terms equal by normalizing (hence the
"shuffler" name). Uses the simplifier for the normalization.
*)
signature Shuffler =
sig
val debug : bool Unsynchronized.ref
val norm_term : theory -> term -> thm
val make_equal : theory -> term -> term -> thm option
val set_prop : theory -> term -> (string * thm) list -> (string * thm) option
val find_potential: theory -> term -> (string * thm) list
val gen_shuffle_tac: Proof.context -> bool -> (string * thm) list -> int -> tactic
val shuffle_tac: Proof.context -> thm list -> int -> tactic
val search_tac : Proof.context -> int -> tactic
val print_shuffles: theory -> unit
val add_shuffle_rule: thm -> theory -> theory
val shuffle_attr: attribute
val setup : theory -> theory
end
structure Shuffler :> Shuffler =
struct
val debug = Unsynchronized.ref false
fun if_debug f x = if !debug then f x else ()
val message = if_debug writeln
val string_of_thm = Print_Mode.setmp [] Display.string_of_thm_without_context;
fun mk_meta_eq th =
(case concl_of th of
Const(@{const_name Trueprop},_) $ (Const(@{const_name HOL.eq},_) $ _ $ _) => th RS eq_reflection
| Const("==",_) $ _ $ _ => th
| _ => raise THM("Not an equality",0,[th]))
handle _ => raise THM("Couldn't make meta equality",0,[th]) (* FIXME avoid handle _ *)
fun mk_obj_eq th =
(case concl_of th of
Const(@{const_name Trueprop},_) $ (Const(@{const_name HOL.eq},_) $ _ $ _) => th
| Const("==",_) $ _ $ _ => th RS meta_eq_to_obj_eq
| _ => raise THM("Not an equality",0,[th]))
handle _ => raise THM("Couldn't make object equality",0,[th]) (* FIXME avoid handle _ *)
structure ShuffleData = Theory_Data
(
type T = thm list
val empty = []
val extend = I
val merge = Thm.merge_thms
)
fun print_shuffles thy =
Pretty.writeln (Pretty.big_list "Shuffle theorems:"
(map (Display.pretty_thm_global thy) (ShuffleData.get thy)))
val weaken =
let
val cert = cterm_of Pure.thy
val P = Free("P",propT)
val Q = Free("Q",propT)
val PQ = Logic.mk_implies(P,Q)
val PPQ = Logic.mk_implies(P,PQ)
val cP = cert P
val cQ = cert Q
val cPQ = cert PQ
val cPPQ = cert PPQ
val th1 = Thm.assume cPQ |> implies_intr_list [cPQ,cP]
val th3 = Thm.assume cP
val th4 = implies_elim_list (Thm.assume cPPQ) [th3,th3]
|> implies_intr_list [cPPQ,cP]
in
Thm.equal_intr th4 th1 |> Drule.export_without_context
end
val imp_comm =
let
val cert = cterm_of Pure.thy
val P = Free("P",propT)
val Q = Free("Q",propT)
val R = Free("R",propT)
val PQR = Logic.mk_implies(P,Logic.mk_implies(Q,R))
val QPR = Logic.mk_implies(Q,Logic.mk_implies(P,R))
val cP = cert P
val cQ = cert Q
val cPQR = cert PQR
val cQPR = cert QPR
val th1 = implies_elim_list (Thm.assume cPQR) [Thm.assume cP,Thm.assume cQ]
|> implies_intr_list [cPQR,cQ,cP]
val th2 = implies_elim_list (Thm.assume cQPR) [Thm.assume cQ,Thm.assume cP]
|> implies_intr_list [cQPR,cP,cQ]
in
Thm.equal_intr th1 th2 |> Drule.export_without_context
end
val def_norm =
let
val cert = cterm_of Pure.thy
val aT = TFree("'a",[])
val bT = TFree("'b",[])
val v = Free("v",aT)
val P = Free("P",aT-->bT)
val Q = Free("Q",aT-->bT)
val cvPQ = cert (list_all ([("v",aT)],Logic.mk_equals(P $ Bound 0,Q $ Bound 0)))
val cPQ = cert (Logic.mk_equals(P,Q))
val cv = cert v
val rew = Thm.assume cvPQ
|> Thm.forall_elim cv
|> Thm.abstract_rule "v" cv
val (lhs,rhs) = Logic.dest_equals(concl_of rew)
val th1 = Thm.transitive (Thm.transitive
(Thm.eta_conversion (cert lhs) |> Thm.symmetric)
rew)
(Thm.eta_conversion (cert rhs))
|> Thm.implies_intr cvPQ
val th2 = Thm.combination (Thm.assume cPQ) (Thm.reflexive cv)
|> Thm.forall_intr cv
|> Thm.implies_intr cPQ
in
Thm.equal_intr th1 th2 |> Drule.export_without_context
end
val all_comm =
let
val cert = cterm_of Pure.thy
val xT = TFree("'a",[])
val yT = TFree("'b",[])
val x = Free("x",xT)
val y = Free("y",yT)
val P = Free("P",xT-->yT-->propT)
val lhs = Logic.all x (Logic.all y (P $ x $ y))
val rhs = Logic.all y (Logic.all x (P $ x $ y))
val cl = cert lhs
val cr = cert rhs
val cx = cert x
val cy = cert y
val th1 = Thm.assume cr
|> forall_elim_list [cy,cx]
|> forall_intr_list [cx,cy]
|> Thm.implies_intr cr
val th2 = Thm.assume cl
|> forall_elim_list [cx,cy]
|> forall_intr_list [cy,cx]
|> Thm.implies_intr cl
in
Thm.equal_intr th1 th2 |> Drule.export_without_context
end
val equiv_comm =
let
val cert = cterm_of Pure.thy
val T = TFree("'a",[])
val t = Free("t",T)
val u = Free("u",T)
val ctu = cert (Logic.mk_equals(t,u))
val cut = cert (Logic.mk_equals(u,t))
val th1 = Thm.assume ctu |> Thm.symmetric |> Thm.implies_intr ctu
val th2 = Thm.assume cut |> Thm.symmetric |> Thm.implies_intr cut
in
Thm.equal_intr th1 th2 |> Drule.export_without_context
end
(* This simplification procedure rewrites !!x y. P x y
deterministicly, in order for the normalization function, defined
below, to handle nested quantifiers robustly *)
local
exception RESULT of int
fun find_bound n (Bound i) = if i = n then raise RESULT 0
else if i = n+1 then raise RESULT 1
else ()
| find_bound n (t $ u) = (find_bound n t; find_bound n u)
| find_bound n (Abs(_,_,t)) = find_bound (n+1) t
| find_bound _ _ = ()
fun swap_bound n (Bound i) = if i = n then Bound (n+1)
else if i = n+1 then Bound n
else Bound i
| swap_bound n (t $ u) = (swap_bound n t $ swap_bound n u)
| swap_bound n (Abs(x,xT,t)) = Abs(x,xT,swap_bound (n+1) t)
| swap_bound n t = t
fun rew_th thy (xv as (x,xT)) (yv as (y,yT)) t =
let
val lhs = list_all ([xv,yv],t)
val rhs = list_all ([yv,xv],swap_bound 0 t)
val rew = Logic.mk_equals (lhs,rhs)
val init = Thm.trivial (cterm_of thy rew)
in
all_comm RS init
end
fun quant_rewrite thy assumes (t as Const("all",T1) $ (Abs(x,xT,Const("all",T2) $ Abs(y,yT,body)))) =
let
val res = (find_bound 0 body;2) handle RESULT i => i
in
case res of
0 => SOME (rew_th thy (x,xT) (y,yT) body)
| 1 => if string_ord(y,x) = LESS
then
let
val newt = Const("all",T1) $ (Abs(y,xT,Const("all",T2) $ Abs(x,yT,body)))
val t_th = Thm.reflexive (cterm_of thy t)
val newt_th = Thm.reflexive (cterm_of thy newt)
in
SOME (Thm.transitive t_th newt_th)
end
else NONE
| _ => error "norm_term (quant_rewrite) internal error"
end
| quant_rewrite _ _ _ = (warning "quant_rewrite: Unknown lhs"; NONE)
fun freeze_thaw_term t =
let
val tvars = OldTerm.term_tvars t
val tfree_names = OldTerm.add_term_tfree_names(t,[])
val (type_inst,_) =
fold (fn (w as (v,_), S) => fn (inst, used) =>
let
val v' = singleton (Name.variant_list used) v
in
((w,TFree(v',S))::inst,v'::used)
end)
tvars ([], tfree_names)
val t' = subst_TVars type_inst t
in
(t', map (fn (w,TFree(v,S)) => (v,TVar(w,S))
| _ => error "Internal error in Shuffler.freeze_thaw") type_inst)
end
fun inst_tfrees thy [] thm = thm
| inst_tfrees thy ((name,U)::rest) thm =
let
val cU = ctyp_of thy U
val tfrees = OldTerm.add_term_tfrees (prop_of thm,[])
val (rens, thm') = Thm.varifyT_global'
(remove (op = o apsnd fst) name tfrees) thm
val mid =
case rens of
[] => thm'
| [((_, S), idx)] => instantiate
([(ctyp_of thy (TVar (idx, S)), cU)], []) thm'
| _ => error "Shuffler.inst_tfrees internal error"
in
inst_tfrees thy rest mid
end
fun is_Abs (Abs _) = true
| is_Abs _ = false
fun eta_redex (t $ Bound 0) =
let
fun free n (Bound i) = i = n
| free n (t $ u) = free n t orelse free n u
| free n (Abs(_,_,t)) = free (n+1) t
| free n _ = false
in
not (free 0 t)
end
| eta_redex _ = false
fun eta_contract thy assumes origt =
let
val (typet,Tinst) = freeze_thaw_term origt
val (init,thaw) = Drule.legacy_freeze_thaw (Thm.reflexive (cterm_of thy typet))
val final = inst_tfrees thy Tinst o thaw
val t = #1 (Logic.dest_equals (prop_of init))
val _ =
let
val lhs = #1 (Logic.dest_equals (prop_of (final init)))
in
if not (lhs aconv origt)
then
writeln (cat_lines
(["Something is utterly wrong: (orig, lhs, frozen type, t, tinst)",
Syntax.string_of_term_global thy origt,
Syntax.string_of_term_global thy lhs,
Syntax.string_of_term_global thy typet,
Syntax.string_of_term_global thy t] @
map (fn (n, T) => n ^ ": " ^ Syntax.string_of_typ_global thy T) Tinst))
else ()
end
in
case t of
Const("all",_) $ (Abs(x,xT,Const("==",eqT) $ P $ Q)) =>
(if eta_redex P andalso eta_redex Q
then
let
val cert = cterm_of thy
val v = Free (singleton (Name.variant_list (Term.add_free_names t [])) "v", xT)
val cv = cert v
val ct = cert t
val th = (Thm.assume ct)
|> Thm.forall_elim cv
|> Thm.abstract_rule x cv
val ext_th = Thm.eta_conversion (cert (Abs(x,xT,P)))
val th' = Thm.transitive (Thm.symmetric ext_th) th
val cu = cert (prop_of th')
val uth = Thm.combination (Thm.assume cu) (Thm.reflexive cv)
val uth' = (Thm.beta_conversion false (cert (Abs(x,xT,Q) $ v)))
|> Thm.transitive uth
|> Thm.forall_intr cv
|> Thm.implies_intr cu
val rew_th = Thm.equal_intr (th' |> Thm.implies_intr ct) uth'
val res = final rew_th
val lhs = (#1 (Logic.dest_equals (prop_of res)))
in
SOME res
end
else NONE)
| _ => NONE
end
fun beta_fun thy assume t =
SOME (Thm.beta_conversion true (cterm_of thy t))
val meta_sym_rew = @{thm refl}
fun equals_fun thy assume t =
case t of
Const("op ==",_) $ u $ v => if Term_Ord.term_ord (u,v) = LESS then SOME (meta_sym_rew) else NONE
| _ => NONE
fun eta_expand thy assumes origt =
let
val (typet,Tinst) = freeze_thaw_term origt
val (init,thaw) = Drule.legacy_freeze_thaw (Thm.reflexive (cterm_of thy typet))
val final = inst_tfrees thy Tinst o thaw
val t = #1 (Logic.dest_equals (prop_of init))
val _ =
let
val lhs = #1 (Logic.dest_equals (prop_of (final init)))
in
if not (lhs aconv origt)
then
writeln (cat_lines
(["Something is utterly wrong: (orig, lhs, frozen type, t, tinst)",
Syntax.string_of_term_global thy origt,
Syntax.string_of_term_global thy lhs,
Syntax.string_of_term_global thy typet,
Syntax.string_of_term_global thy t] @
map (fn (n, T) => n ^ ": " ^ Syntax.string_of_typ_global thy T) Tinst))
else ()
end
in
case t of
Const("==",T) $ P $ Q =>
if is_Abs P orelse is_Abs Q
then (case domain_type T of
Type("fun",[aT,bT]) =>
let
val cert = cterm_of thy
val vname = singleton (Name.variant_list (Term.add_free_names t [])) "v"
val v = Free(vname,aT)
val cv = cert v
val ct = cert t
val th1 = (Thm.combination (Thm.assume ct) (Thm.reflexive cv))
|> Thm.forall_intr cv
|> Thm.implies_intr ct
val concl = cert (concl_of th1)
val th2 = (Thm.assume concl)
|> Thm.forall_elim cv
|> Thm.abstract_rule vname cv
val (lhs,rhs) = Logic.dest_equals (prop_of th2)
val elhs = Thm.eta_conversion (cert lhs)
val erhs = Thm.eta_conversion (cert rhs)
val th2' = Thm.transitive
(Thm.transitive (Thm.symmetric elhs) th2)
erhs
val res = Thm.equal_intr th1 (th2' |> Thm.implies_intr concl)
val res' = final res
in
SOME res'
end
| _ => NONE)
else NONE
| _ => error ("Bad eta_expand argument" ^ Syntax.string_of_term_global thy t)
end;
fun mk_tfree s = TFree("'"^s,[])
fun mk_free s t = Free (s,t)
val xT = mk_tfree "a"
val yT = mk_tfree "b"
val x = Free ("x", xT)
val y = Free ("y", yT)
val P = mk_free "P" (xT-->yT-->propT)
val Q = mk_free "Q" (xT-->yT)
val R = mk_free "R" (xT-->yT)
val S = mk_free "S" xT
val S' = mk_free "S'" xT
in
fun beta_simproc thy = Simplifier.simproc_global_i
thy
"Beta-contraction"
[Abs("x",xT,Q) $ S]
beta_fun
fun equals_simproc thy = Simplifier.simproc_global_i
thy
"Ordered rewriting of meta equalities"
[Const("op ==",xT) $ S $ S']
equals_fun
fun quant_simproc thy = Simplifier.simproc_global_i
thy
"Ordered rewriting of nested quantifiers"
[Logic.all x (Logic.all y (P $ x $ y))]
quant_rewrite
fun eta_expand_simproc thy = Simplifier.simproc_global_i
thy
"Smart eta-expansion by equivalences"
[Logic.mk_equals(Q,R)]
eta_expand
fun eta_contract_simproc thy = Simplifier.simproc_global_i
thy
"Smart handling of eta-contractions"
[Logic.all x (Logic.mk_equals (Q $ x, R $ x))]
eta_contract
end
(* Disambiguates the names of bound variables in a term, returning t
== t' where all the names of bound variables in t' are unique *)
fun disamb_bound thy t =
let
fun F (t $ u,idx) =
let
val (t',idx') = F (t,idx)
val (u',idx'') = F (u,idx')
in
(t' $ u',idx'')
end
| F (Abs(x,xT,t),idx) =
let
val x' = "x" ^ string_of_int idx
val (t',idx') = F (t,idx+1)
in
(Abs(x',xT,t'),idx')
end
| F arg = arg
val (t',_) = F (t,0)
val ct = cterm_of thy t
val ct' = cterm_of thy t'
val res = Thm.transitive (Thm.reflexive ct) (Thm.reflexive ct')
val _ = message ("disamb_term: " ^ (string_of_thm res))
in
res
end
(* Transforms a term t to some normal form t', returning the theorem t
== t'. This is originally a help function for make_equal, but might
be handy in its own right, for example for indexing terms. *)
fun norm_term thy t =
let
val norms = ShuffleData.get thy
val ss = Simplifier.global_context thy empty_ss
setmksimps (K single)
addsimps (map (Thm.transfer thy) norms)
addsimprocs [quant_simproc thy, eta_expand_simproc thy,eta_contract_simproc thy]
fun chain f th =
let
val rhs = Thm.rhs_of th
in
Thm.transitive th (f rhs)
end
val th =
t |> disamb_bound thy
|> chain (Simplifier.full_rewrite ss)
|> chain Thm.eta_conversion
|> Thm.strip_shyps
val _ = message ("norm_term: " ^ (string_of_thm th))
in
th
end
(* Closes a theorem with respect to free and schematic variables (does
not touch type variables, though). *)
fun close_thm th =
let
val thy = Thm.theory_of_thm th
val c = prop_of th
val vars = OldTerm.add_term_frees (c, OldTerm.add_term_vars(c,[]))
in
Drule.forall_intr_list (map (cterm_of thy) vars) th
end
(* Normalizes a theorem's conclusion using norm_term. *)
fun norm_thm thy th =
let
val c = prop_of th
in
Thm.equal_elim (norm_term thy c) th
end
(* make_equal thy t u tries to construct the theorem t == u under the
signature thy. If it succeeds, SOME (t == u) is returned, otherwise
NONE is returned. *)
fun make_equal thy t u =
let
val t_is_t' = norm_term thy t
val u_is_u' = norm_term thy u
val th = Thm.transitive t_is_t' (Thm.symmetric u_is_u')
val _ = message ("make_equal: SOME " ^ (string_of_thm th))
in
SOME th
end
handle e as THM _ => (message "make_equal: NONE";NONE)
fun match_consts ignore t (* th *) =
let
fun add_consts (Const (c, _), cs) =
if member (op =) ignore c
then cs
else insert (op =) c cs
| add_consts (t $ u, cs) = add_consts (t, add_consts (u, cs))
| add_consts (Abs (_, _, t), cs) = add_consts (t, cs)
| add_consts (_, cs) = cs
val t_consts = add_consts(t,[])
in
fn (name,th) =>
let
val th_consts = add_consts(prop_of th,[])
in
eq_set (op =) (t_consts, th_consts)
end
end
val collect_ignored = fold_rev (fn thm => fn cs =>
let
val (lhs, rhs) = Logic.dest_equals (prop_of thm);
val consts_lhs = Term.add_const_names lhs [];
val consts_rhs = Term.add_const_names rhs [];
val ignore_lhs = subtract (op =) consts_rhs consts_lhs;
val ignore_rhs = subtract (op =) consts_lhs consts_rhs;
in
fold_rev (insert (op =)) cs (ignore_lhs @ ignore_rhs)
end)
(* set_prop t thms tries to make a theorem with the proposition t from
one of the theorems thms, by shuffling the propositions around. If it
succeeds, SOME theorem is returned, otherwise NONE. *)
fun set_prop thy t =
let
val vars = OldTerm.add_term_frees (t, OldTerm.add_term_vars (t,[]))
val closed_t = fold_rev Logic.all vars t
val rew_th = norm_term thy closed_t
val rhs = Thm.rhs_of rew_th
val shuffles = ShuffleData.get thy
fun process [] = NONE
| process ((name,th)::thms) =
let
val norm_th = Thm.varifyT_global (norm_thm thy (close_thm (Thm.transfer thy th)))
val triv_th = Thm.trivial rhs
val _ = message ("Shuffler.set_prop: Gluing together " ^ (string_of_thm norm_th) ^ " and " ^ (string_of_thm triv_th))
val mod_th = case Seq.pull (Thm.bicompose false (*true*) (false,norm_th,0) 1 triv_th) of
SOME(th,_) => SOME th
| NONE => NONE
in
case mod_th of
SOME mod_th =>
let
val closed_th = Thm.equal_elim (Thm.symmetric rew_th) mod_th
in
message ("Shuffler.set_prop succeeded by " ^ name);
SOME (name,forall_elim_list (map (cterm_of thy) vars) closed_th)
end
| NONE => process thms
end
handle e as THM _ => process thms
in
fn thms =>
case process thms of
res as SOME (name,th) => if (prop_of th) aconv t
then res
else error "Internal error in set_prop"
| NONE => NONE
end
fun find_potential thy t =
let
val shuffles = ShuffleData.get thy
val ignored = collect_ignored shuffles []
val all_thms =
map (`Thm.get_name_hint) (maps #2 (Facts.dest_static [] (Global_Theory.facts_of thy)))
in
filter (match_consts ignored t) all_thms
end
fun gen_shuffle_tac ctxt search thms = SUBGOAL (fn (t, i) =>
let
val thy = Proof_Context.theory_of ctxt
val set = set_prop thy t
fun process_tac thms st =
case set thms of
SOME (_,th) => Seq.of_list (compose (th,i,st))
| NONE => Seq.empty
in
process_tac thms APPEND
(if search then process_tac (find_potential thy t) else no_tac)
end)
fun shuffle_tac ctxt thms =
gen_shuffle_tac ctxt false (map (pair "") thms);
fun search_tac ctxt =
gen_shuffle_tac ctxt true (map (pair "premise") (Assumption.all_prems_of ctxt));
fun add_shuffle_rule thm thy =
let
val shuffles = ShuffleData.get thy
in
if exists (curry Thm.eq_thm thm) shuffles
then (warning ((string_of_thm thm) ^ " already known to the shuffler");
thy)
else ShuffleData.put (thm::shuffles) thy
end
val shuffle_attr = Thm.declaration_attribute (fn th => Context.mapping (add_shuffle_rule th) I);
val setup =
Method.setup @{binding shuffle_tac}
(Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (shuffle_tac ctxt ths)))
"solve goal by shuffling terms around" #>
Method.setup @{binding search_tac}
(Scan.succeed (SIMPLE_METHOD' o search_tac)) "search for suitable theorems" #>
add_shuffle_rule weaken #>
add_shuffle_rule equiv_comm #>
add_shuffle_rule imp_comm #>
add_shuffle_rule Drule.norm_hhf_eq #>
add_shuffle_rule Drule.triv_forall_equality #>
Attrib.setup @{binding shuffle_rule} (Scan.succeed shuffle_attr) "declare rule for shuffler";
end