(* Title: Pure/drule.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Derived rules and other operations on theorems.
*)
infix 0 RS RSN RL RLN MRS OF COMP INCR_COMP COMP_INCR;
signature BASIC_DRULE =
sig
val mk_implies: cterm * cterm -> cterm
val list_implies: cterm list * cterm -> cterm
val strip_imp_prems: cterm -> cterm list
val strip_imp_concl: cterm -> cterm
val cprems_of: thm -> cterm list
val cterm_fun: (term -> term) -> (cterm -> cterm)
val ctyp_fun: (typ -> typ) -> (ctyp -> ctyp)
val forall_intr_list: cterm list -> thm -> thm
val forall_intr_vars: thm -> thm
val forall_elim_list: cterm list -> thm -> thm
val gen_all: thm -> thm
val lift_all: cterm -> thm -> thm
val implies_elim_list: thm -> thm list -> thm
val implies_intr_list: cterm list -> thm -> thm
val instantiate_normalize: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm
val zero_var_indexes_list: thm list -> thm list
val zero_var_indexes: thm -> thm
val implies_intr_hyps: thm -> thm
val rotate_prems: int -> thm -> thm
val rearrange_prems: int list -> thm -> thm
val RSN: thm * (int * thm) -> thm
val RS: thm * thm -> thm
val RLN: thm list * (int * thm list) -> thm list
val RL: thm list * thm list -> thm list
val MRS: thm list * thm -> thm
val OF: thm * thm list -> thm
val COMP: thm * thm -> thm
val INCR_COMP: thm * thm -> thm
val COMP_INCR: thm * thm -> thm
val cterm_instantiate: (cterm * cterm) list -> thm -> thm
val size_of_thm: thm -> int
val reflexive_thm: thm
val symmetric_thm: thm
val transitive_thm: thm
val extensional: thm -> thm
val asm_rl: thm
val cut_rl: thm
val revcut_rl: thm
val thin_rl: thm
val instantiate': ctyp option list -> cterm option list -> thm -> thm
end;
signature DRULE =
sig
include BASIC_DRULE
val generalize: string list * string list -> thm -> thm
val list_comb: cterm * cterm list -> cterm
val strip_comb: cterm -> cterm * cterm list
val strip_type: ctyp -> ctyp list * ctyp
val beta_conv: cterm -> cterm -> cterm
val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
val flexflex_unique: thm -> thm
val export_without_context: thm -> thm
val export_without_context_open: thm -> thm
val store_thm: binding -> thm -> thm
val store_standard_thm: binding -> thm -> thm
val store_thm_open: binding -> thm -> thm
val store_standard_thm_open: binding -> thm -> thm
val multi_resolve: thm list -> thm -> thm Seq.seq
val multi_resolves: thm list -> thm list -> thm Seq.seq
val compose: thm * int * thm -> thm
val equals_cong: thm
val imp_cong: thm
val swap_prems_eq: thm
val imp_cong_rule: thm -> thm -> thm
val arg_cong_rule: cterm -> thm -> thm
val binop_cong_rule: cterm -> thm -> thm -> thm
val fun_cong_rule: thm -> cterm -> thm
val beta_eta_conversion: cterm -> thm
val eta_long_conversion: cterm -> thm
val eta_contraction_rule: thm -> thm
val norm_hhf_eq: thm
val norm_hhf_eqs: thm list
val is_norm_hhf: term -> bool
val norm_hhf: theory -> term -> term
val norm_hhf_cterm: cterm -> cterm
val protect: cterm -> cterm
val protectI: thm
val protectD: thm
val protect_cong: thm
val implies_intr_protected: cterm list -> thm -> thm
val termI: thm
val mk_term: cterm -> thm
val dest_term: thm -> cterm
val cterm_rule: (thm -> thm) -> cterm -> cterm
val dummy_thm: thm
val sort_constraintI: thm
val sort_constraint_eq: thm
val with_subgoal: int -> (thm -> thm) -> thm -> thm
val comp_no_flatten: thm * int -> int -> thm -> thm
val rename_bvars: (string * string) list -> thm -> thm
val rename_bvars': string option list -> thm -> thm
val incr_indexes: thm -> thm -> thm
val incr_indexes2: thm -> thm -> thm -> thm
val triv_forall_equality: thm
val distinct_prems_rl: thm
val equal_intr_rule: thm
val equal_elim_rule1: thm
val equal_elim_rule2: thm
val remdups_rl: thm
val abs_def: thm -> thm
end;
structure Drule: DRULE =
struct
(** some cterm->cterm operations: faster than calling cterm_of! **)
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
fun strip_imp_prems ct =
let val (cA, cB) = Thm.dest_implies ct
in cA :: strip_imp_prems cB end
handle TERM _ => [];
(* A1==>...An==>B goes to B, where B is not an implication *)
fun strip_imp_concl ct =
(case Thm.term_of ct of
Const ("==>", _) $ _ $ _ => strip_imp_concl (Thm.dest_arg ct)
| _ => ct);
(*The premises of a theorem, as a cterm list*)
val cprems_of = strip_imp_prems o cprop_of;
fun cterm_fun f ct = Thm.cterm_of (Thm.theory_of_cterm ct) (f (Thm.term_of ct));
fun ctyp_fun f cT = Thm.ctyp_of (Thm.theory_of_ctyp cT) (f (Thm.typ_of cT));
fun certify t = Thm.cterm_of (Context.the_theory (Context.the_thread_data ())) t;
val implies = certify Logic.implies;
fun mk_implies (A, B) = Thm.apply (Thm.apply implies A) B;
(*cterm version of list_implies: [A1,...,An], B goes to [|A1;==>;An|]==>B *)
fun list_implies([], B) = B
| list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));
(*cterm version of list_comb: maps (f, [t1,...,tn]) to f(t1,...,tn) *)
fun list_comb (f, []) = f
| list_comb (f, t::ts) = list_comb (Thm.apply f t, ts);
(*cterm version of strip_comb: maps f(t1,...,tn) to (f, [t1,...,tn]) *)
fun strip_comb ct =
let
fun stripc (p as (ct, cts)) =
let val (ct1, ct2) = Thm.dest_comb ct
in stripc (ct1, ct2 :: cts) end handle CTERM _ => p
in stripc (ct, []) end;
(* cterm version of strip_type: maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T) *)
fun strip_type cT = (case Thm.typ_of cT of
Type ("fun", _) =>
let
val [cT1, cT2] = Thm.dest_ctyp cT;
val (cTs, cT') = strip_type cT2
in (cT1 :: cTs, cT') end
| _ => ([], cT));
(*Beta-conversion for cterms, where x is an abstraction. Simply returns the rhs
of the meta-equality returned by the beta_conversion rule.*)
fun beta_conv x y =
Thm.dest_arg (cprop_of (Thm.beta_conversion false (Thm.apply x y)));
(*** Find the type (sort) associated with a (T)Var or (T)Free in a term
Used for establishing default types (of variables) and sorts (of
type variables) when reading another term.
Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
***)
fun types_sorts thm =
let
val vars = Thm.fold_terms Term.add_vars thm [];
val frees = Thm.fold_terms Term.add_frees thm [];
val tvars = Thm.fold_terms Term.add_tvars thm [];
val tfrees = Thm.fold_terms Term.add_tfrees thm [];
fun types (a, i) =
if i < 0 then AList.lookup (op =) frees a else AList.lookup (op =) vars (a, i);
fun sorts (a, i) =
if i < 0 then AList.lookup (op =) tfrees a else AList.lookup (op =) tvars (a, i);
in (types, sorts) end;
(** Standardization of rules **)
(*Generalization over a list of variables*)
val forall_intr_list = fold_rev Thm.forall_intr;
(*Generalization over Vars -- canonical order*)
fun forall_intr_vars th =
fold Thm.forall_intr
(map (Thm.cterm_of (Thm.theory_of_thm th) o Var) (Thm.fold_terms Term.add_vars th [])) th;
fun outer_params t =
let val vs = Term.strip_all_vars t
in Name.variant_list [] (map (Name.clean o #1) vs) ~~ map #2 vs end;
(*generalize outermost parameters*)
fun gen_all th =
let
val thy = Thm.theory_of_thm th;
val {prop, maxidx, ...} = Thm.rep_thm th;
val cert = Thm.cterm_of thy;
fun elim (x, T) = Thm.forall_elim (cert (Var ((x, maxidx + 1), T)));
in fold elim (outer_params prop) th end;
(*lift vars wrt. outermost goal parameters
-- reverses the effect of gen_all modulo higher-order unification*)
fun lift_all goal th =
let
val thy = Theory.merge (Thm.theory_of_cterm goal, Thm.theory_of_thm th);
val cert = Thm.cterm_of thy;
val maxidx = Thm.maxidx_of th;
val ps = outer_params (Thm.term_of goal)
|> map (fn (x, T) => Var ((x, maxidx + 1), Logic.incr_tvar (maxidx + 1) T));
val Ts = map Term.fastype_of ps;
val inst = Thm.fold_terms Term.add_vars th [] |> map (fn (xi, T) =>
(cert (Var (xi, T)), cert (Term.list_comb (Var (xi, Ts ---> T), ps))));
in
th |> Thm.instantiate ([], inst)
|> fold_rev (Thm.forall_intr o cert) ps
end;
(*direct generalization*)
fun generalize names th = Thm.generalize names (Thm.maxidx_of th + 1) th;
(*specialization over a list of cterms*)
val forall_elim_list = fold Thm.forall_elim;
(*maps A1,...,An |- B to [| A1;...;An |] ==> B*)
val implies_intr_list = fold_rev Thm.implies_intr;
(*maps [| A1;...;An |] ==> B and [A1,...,An] to B*)
fun implies_elim_list impth ths = fold Thm.elim_implies ths impth;
(*Reset Var indexes to zero, renaming to preserve distinctness*)
fun zero_var_indexes_list [] = []
| zero_var_indexes_list ths =
let
val thy = Theory.merge_list (map Thm.theory_of_thm ths);
val certT = Thm.ctyp_of thy and cert = Thm.cterm_of thy;
val (instT, inst) = Term_Subst.zero_var_indexes_inst (map Thm.full_prop_of ths);
val cinstT = map (fn (v, T) => (certT (TVar v), certT T)) instT;
val cinst = map (fn (v, t) => (cert (Var v), cert t)) inst;
in map (Thm.adjust_maxidx_thm ~1 o Thm.instantiate (cinstT, cinst)) ths end;
val zero_var_indexes = singleton zero_var_indexes_list;
(** Standard form of object-rule: no hypotheses, flexflex constraints,
Frees, or outer quantifiers; all generality expressed by Vars of index 0.**)
(*Discharge all hypotheses.*)
fun implies_intr_hyps th =
fold Thm.implies_intr (#hyps (Thm.crep_thm th)) th;
(*Squash a theorem's flexflex constraints provided it can be done uniquely.
This step can lose information.*)
fun flexflex_unique th =
if null (Thm.tpairs_of th) then th else
case distinct Thm.eq_thm (Seq.list_of (Thm.flexflex_rule th)) of
[th] => th
| [] => raise THM("flexflex_unique: impossible constraints", 0, [th])
| _ => raise THM("flexflex_unique: multiple unifiers", 0, [th]);
(* old-style export without context *)
val export_without_context_open =
implies_intr_hyps
#> Thm.forall_intr_frees
#> `Thm.maxidx_of
#-> (fn maxidx =>
Thm.forall_elim_vars (maxidx + 1)
#> Thm.strip_shyps
#> zero_var_indexes
#> Thm.varifyT_global);
val export_without_context =
flexflex_unique
#> export_without_context_open
#> Thm.close_derivation;
(*Rotates a rule's premises to the left by k*)
fun rotate_prems 0 = I
| rotate_prems k = Thm.permute_prems 0 k;
fun with_subgoal i f = rotate_prems (i - 1) #> f #> rotate_prems (1 - i);
(*Permute prems, where the i-th position in the argument list (counting from 0)
gives the position within the original thm to be transferred to position i.
Any remaining trailing positions are left unchanged.*)
val rearrange_prems =
let
fun rearr new [] thm = thm
| rearr new (p :: ps) thm =
rearr (new + 1)
(map (fn q => if new <= q andalso q < p then q + 1 else q) ps)
(Thm.permute_prems (new + 1) (new - p) (Thm.permute_prems new (p - new) thm))
in rearr 0 end;
(*Resolution: multiple arguments, multiple results*)
local
fun res th i rule =
Thm.biresolution false [(false, th)] i rule handle THM _ => Seq.empty;
fun multi_res _ [] rule = Seq.single rule
| multi_res i (th :: ths) rule = Seq.maps (res th i) (multi_res (i + 1) ths rule);
in
val multi_resolve = multi_res 1;
fun multi_resolves facts rules = Seq.maps (multi_resolve facts) (Seq.of_list rules);
end;
(*Resolution: exactly one resolvent must be produced*)
fun tha RSN (i, thb) =
(case Seq.chop 2 (Thm.biresolution false [(false, tha)] i thb) of
([th], _) => th
| ([], _) => raise THM ("RSN: no unifiers", i, [tha, thb])
| _ => raise THM ("RSN: multiple unifiers", i, [tha, thb]));
(*Resolution: P==>Q, Q==>R gives P==>R*)
fun tha RS thb = tha RSN (1,thb);
(*For joining lists of rules*)
fun thas RLN (i, thbs) =
let val resolve = Thm.biresolution false (map (pair false) thas) i
fun resb thb = Seq.list_of (resolve thb) handle THM _ => []
in maps resb thbs end;
fun thas RL thbs = thas RLN (1, thbs);
(*Isar-style multi-resolution*)
fun bottom_rl OF rls =
(case Seq.chop 2 (multi_resolve rls bottom_rl) of
([th], _) => th
| ([], _) => raise THM ("OF: no unifiers", 0, bottom_rl :: rls)
| _ => raise THM ("OF: multiple unifiers", 0, bottom_rl :: rls));
(*Resolve a list of rules against bottom_rl from right to left;
makes proof trees*)
fun rls MRS bottom_rl = bottom_rl OF rls;
(*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R
with no lifting or renaming! Q may contain ==> or meta-quants
ALWAYS deletes premise i *)
fun compose (tha, i, thb) =
Thm.bicompose {flatten = true, match = false, incremented = false} (false, tha, 0) i thb
|> Seq.list_of |> distinct Thm.eq_thm
|> (fn [th] => th | _ => raise THM ("compose: unique result expected", i, [tha, thb]));
(** theorem equality **)
(*Useful "distance" function for BEST_FIRST*)
val size_of_thm = size_of_term o Thm.full_prop_of;
(*** Meta-Rewriting Rules ***)
val read_prop = certify o Simple_Syntax.read_prop;
fun store_thm name th =
Context.>>> (Context.map_theory_result (Global_Theory.store_thm (name, th)));
fun store_thm_open name th =
Context.>>> (Context.map_theory_result (Global_Theory.store_thm_open (name, th)));
fun store_standard_thm name th = store_thm name (export_without_context th);
fun store_standard_thm_open name thm = store_thm_open name (export_without_context_open thm);
val reflexive_thm =
let val cx = certify (Var(("x",0),TVar(("'a",0),[])))
in store_standard_thm_open (Binding.name "reflexive") (Thm.reflexive cx) end;
val symmetric_thm =
let
val xy = read_prop "x::'a == y::'a";
val thm = Thm.implies_intr xy (Thm.symmetric (Thm.assume xy));
in store_standard_thm_open (Binding.name "symmetric") thm end;
val transitive_thm =
let
val xy = read_prop "x::'a == y::'a";
val yz = read_prop "y::'a == z::'a";
val xythm = Thm.assume xy;
val yzthm = Thm.assume yz;
val thm = Thm.implies_intr yz (Thm.transitive xythm yzthm);
in store_standard_thm_open (Binding.name "transitive") thm end;
fun extensional eq =
let val eq' =
Thm.abstract_rule "x" (Thm.dest_arg (fst (Thm.dest_equals (cprop_of eq)))) eq
in Thm.equal_elim (Thm.eta_conversion (cprop_of eq')) eq' end;
val equals_cong =
store_standard_thm_open (Binding.name "equals_cong")
(Thm.reflexive (read_prop "x::'a == y::'a"));
val imp_cong =
let
val ABC = read_prop "A ==> B::prop == C::prop"
val AB = read_prop "A ==> B"
val AC = read_prop "A ==> C"
val A = read_prop "A"
in
store_standard_thm_open (Binding.name "imp_cong") (Thm.implies_intr ABC (Thm.equal_intr
(Thm.implies_intr AB (Thm.implies_intr A
(Thm.equal_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A))
(Thm.implies_elim (Thm.assume AB) (Thm.assume A)))))
(Thm.implies_intr AC (Thm.implies_intr A
(Thm.equal_elim (Thm.symmetric (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)))
(Thm.implies_elim (Thm.assume AC) (Thm.assume A)))))))
end;
val swap_prems_eq =
let
val ABC = read_prop "A ==> B ==> C"
val BAC = read_prop "B ==> A ==> C"
val A = read_prop "A"
val B = read_prop "B"
in
store_standard_thm_open (Binding.name "swap_prems_eq")
(Thm.equal_intr
(Thm.implies_intr ABC (Thm.implies_intr B (Thm.implies_intr A
(Thm.implies_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)) (Thm.assume B)))))
(Thm.implies_intr BAC (Thm.implies_intr A (Thm.implies_intr B
(Thm.implies_elim (Thm.implies_elim (Thm.assume BAC) (Thm.assume B)) (Thm.assume A))))))
end;
val imp_cong_rule = Thm.combination o Thm.combination (Thm.reflexive implies);
fun arg_cong_rule ct th = Thm.combination (Thm.reflexive ct) th; (*AP_TERM in LCF/HOL*)
fun fun_cong_rule th ct = Thm.combination th (Thm.reflexive ct); (*AP_THM in LCF/HOL*)
fun binop_cong_rule ct th1 th2 = Thm.combination (arg_cong_rule ct th1) th2;
local
val dest_eq = Thm.dest_equals o cprop_of
val rhs_of = snd o dest_eq
in
fun beta_eta_conversion t =
let val thm = Thm.beta_conversion true t
in Thm.transitive thm (Thm.eta_conversion (rhs_of thm)) end
end;
fun eta_long_conversion ct =
Thm.transitive
(beta_eta_conversion ct)
(Thm.symmetric (beta_eta_conversion (cterm_fun (Envir.eta_long []) ct)));
(*Contract all eta-redexes in the theorem, lest they give rise to needless abstractions*)
fun eta_contraction_rule th =
Thm.equal_elim (Thm.eta_conversion (cprop_of th)) th;
(* abs_def *)
(*
f ?x1 ... ?xn == u
--------------------
f == %x1 ... xn. u
*)
local
fun contract_lhs th =
Thm.transitive (Thm.symmetric (beta_eta_conversion
(fst (Thm.dest_equals (cprop_of th))))) th;
fun var_args ct =
(case try Thm.dest_comb ct of
SOME (f, arg) =>
(case Thm.term_of arg of
Var ((x, _), _) => update (eq_snd (op aconvc)) (x, arg) (var_args f)
| _ => [])
| NONE => []);
in
fun abs_def th =
let
val th' = contract_lhs th;
val args = var_args (Thm.lhs_of th');
in contract_lhs (fold (uncurry Thm.abstract_rule) args th') end;
end;
(*** Some useful meta-theorems ***)
(*The rule V/V, obtains assumption solving for eresolve_tac*)
val asm_rl = store_standard_thm_open (Binding.name "asm_rl") (Thm.trivial (read_prop "?psi"));
(*Meta-level cut rule: [| V==>W; V |] ==> W *)
val cut_rl =
store_standard_thm_open (Binding.name "cut_rl")
(Thm.trivial (read_prop "?psi ==> ?theta"));
(*Generalized elim rule for one conclusion; cut_rl with reversed premises:
[| PROP V; PROP V ==> PROP W |] ==> PROP W *)
val revcut_rl =
let
val V = read_prop "V";
val VW = read_prop "V ==> W";
in
store_standard_thm_open (Binding.name "revcut_rl")
(Thm.implies_intr V (Thm.implies_intr VW (Thm.implies_elim (Thm.assume VW) (Thm.assume V))))
end;
(*for deleting an unwanted assumption*)
val thin_rl =
let
val V = read_prop "V";
val W = read_prop "W";
val thm = Thm.implies_intr V (Thm.implies_intr W (Thm.assume W));
in store_standard_thm_open (Binding.name "thin_rl") thm end;
(* (!!x. PROP ?V) == PROP ?V Allows removal of redundant parameters*)
val triv_forall_equality =
let
val V = read_prop "V";
val QV = read_prop "!!x::'a. V";
val x = certify (Free ("x", Term.aT []));
in
store_standard_thm_open (Binding.name "triv_forall_equality")
(Thm.equal_intr (Thm.implies_intr QV (Thm.forall_elim x (Thm.assume QV)))
(Thm.implies_intr V (Thm.forall_intr x (Thm.assume V))))
end;
(* (PROP ?Phi ==> PROP ?Phi ==> PROP ?Psi) ==>
(PROP ?Phi ==> PROP ?Psi)
*)
val distinct_prems_rl =
let
val AAB = read_prop "Phi ==> Phi ==> Psi";
val A = read_prop "Phi";
in
store_standard_thm_open (Binding.name "distinct_prems_rl")
(implies_intr_list [AAB, A] (implies_elim_list (Thm.assume AAB) [Thm.assume A, Thm.assume A]))
end;
(* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]
==> PROP ?phi == PROP ?psi
Introduction rule for == as a meta-theorem.
*)
val equal_intr_rule =
let
val PQ = read_prop "phi ==> psi";
val QP = read_prop "psi ==> phi";
in
store_standard_thm_open (Binding.name "equal_intr_rule")
(Thm.implies_intr PQ (Thm.implies_intr QP (Thm.equal_intr (Thm.assume PQ) (Thm.assume QP))))
end;
(* PROP ?phi == PROP ?psi ==> PROP ?phi ==> PROP ?psi *)
val equal_elim_rule1 =
let
val eq = read_prop "phi::prop == psi::prop";
val P = read_prop "phi";
in
store_standard_thm_open (Binding.name "equal_elim_rule1")
(Thm.equal_elim (Thm.assume eq) (Thm.assume P) |> implies_intr_list [eq, P])
end;
(* PROP ?psi == PROP ?phi ==> PROP ?phi ==> PROP ?psi *)
val equal_elim_rule2 =
store_standard_thm_open (Binding.name "equal_elim_rule2")
(symmetric_thm RS equal_elim_rule1);
(* PROP ?phi ==> PROP ?phi ==> PROP ?psi ==> PROP ?psi *)
val remdups_rl =
let
val P = read_prop "phi";
val Q = read_prop "psi";
val thm = implies_intr_list [P, P, Q] (Thm.assume Q);
in store_standard_thm_open (Binding.name "remdups_rl") thm end;
(** embedded terms and types **)
local
val A = certify (Free ("A", propT));
val axiom = Thm.unvarify_global o Thm.axiom (Context.the_theory (Context.the_thread_data ()));
val prop_def = axiom "Pure.prop_def";
val term_def = axiom "Pure.term_def";
val sort_constraint_def = axiom "Pure.sort_constraint_def";
val C = Thm.lhs_of sort_constraint_def;
val T = Thm.dest_arg C;
val CA = mk_implies (C, A);
in
(* protect *)
val protect = Thm.apply (certify Logic.protectC);
val protectI =
store_standard_thm (Binding.conceal (Binding.name "protectI"))
(Thm.equal_elim (Thm.symmetric prop_def) (Thm.assume A));
val protectD =
store_standard_thm (Binding.conceal (Binding.name "protectD"))
(Thm.equal_elim prop_def (Thm.assume (protect A)));
val protect_cong =
store_standard_thm_open (Binding.name "protect_cong") (Thm.reflexive (protect A));
fun implies_intr_protected asms th =
let val asms' = map protect asms in
implies_elim_list
(implies_intr_list asms th)
(map (fn asm' => Thm.assume asm' RS protectD) asms')
|> implies_intr_list asms'
end;
(* term *)
val termI =
store_standard_thm (Binding.conceal (Binding.name "termI"))
(Thm.equal_elim (Thm.symmetric term_def) (Thm.forall_intr A (Thm.trivial A)));
fun mk_term ct =
let
val thy = Thm.theory_of_cterm ct;
val cert = Thm.cterm_of thy;
val certT = Thm.ctyp_of thy;
val T = Thm.typ_of (Thm.ctyp_of_term ct);
val a = certT (TVar (("'a", 0), []));
val x = cert (Var (("x", 0), T));
in Thm.instantiate ([(a, certT T)], [(x, ct)]) termI end;
fun dest_term th =
let val cprop = strip_imp_concl (Thm.cprop_of th) in
if can Logic.dest_term (Thm.term_of cprop) then
Thm.dest_arg cprop
else raise THM ("dest_term", 0, [th])
end;
fun cterm_rule f = dest_term o f o mk_term;
val dummy_thm = mk_term (certify Term.dummy_prop);
(* sort_constraint *)
val sort_constraintI =
store_standard_thm (Binding.conceal (Binding.name "sort_constraintI"))
(Thm.equal_elim (Thm.symmetric sort_constraint_def) (mk_term T));
val sort_constraint_eq =
store_standard_thm (Binding.conceal (Binding.name "sort_constraint_eq"))
(Thm.equal_intr
(Thm.implies_intr CA (Thm.implies_elim (Thm.assume CA)
(Thm.unvarify_global sort_constraintI)))
(implies_intr_list [A, C] (Thm.assume A)));
end;
(* HHF normalization *)
(* (PROP ?phi ==> (!!x. PROP ?psi x)) == (!!x. PROP ?phi ==> PROP ?psi x) *)
val norm_hhf_eq =
let
val aT = TFree ("'a", []);
val x = Free ("x", aT);
val phi = Free ("phi", propT);
val psi = Free ("psi", aT --> propT);
val cx = certify x;
val cphi = certify phi;
val lhs = certify (Logic.mk_implies (phi, Logic.all x (psi $ x)));
val rhs = certify (Logic.all x (Logic.mk_implies (phi, psi $ x)));
in
Thm.equal_intr
(Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)
|> Thm.forall_elim cx
|> Thm.implies_intr cphi
|> Thm.forall_intr cx
|> Thm.implies_intr lhs)
(Thm.implies_elim
(Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)
|> Thm.forall_intr cx
|> Thm.implies_intr cphi
|> Thm.implies_intr rhs)
|> store_standard_thm_open (Binding.name "norm_hhf_eq")
end;
val norm_hhf_prop = Logic.dest_equals (Thm.prop_of norm_hhf_eq);
val norm_hhf_eqs = [norm_hhf_eq, sort_constraint_eq];
fun is_norm_hhf (Const ("Pure.sort_constraint", _)) = false
| is_norm_hhf (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false
| is_norm_hhf (Abs _ $ _) = false
| is_norm_hhf (t $ u) = is_norm_hhf t andalso is_norm_hhf u
| is_norm_hhf (Abs (_, _, t)) = is_norm_hhf t
| is_norm_hhf _ = true;
fun norm_hhf thy t =
if is_norm_hhf t then t
else Pattern.rewrite_term thy [norm_hhf_prop] [] t;
fun norm_hhf_cterm ct =
if is_norm_hhf (Thm.term_of ct) then ct
else cterm_fun (Pattern.rewrite_term (Thm.theory_of_cterm ct) [norm_hhf_prop] []) ct;
(* var indexes *)
fun incr_indexes th = Thm.incr_indexes (Thm.maxidx_of th + 1);
fun incr_indexes2 th1 th2 =
Thm.incr_indexes (Int.max (Thm.maxidx_of th1, Thm.maxidx_of th2) + 1);
local
(*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
fun comp incremented th1 th2 =
Thm.bicompose {flatten = true, match = false, incremented = incremented} (false, th1, 0) 1 th2
|> Seq.list_of |> distinct Thm.eq_thm
|> (fn [th] => th | _ => raise THM ("COMP", 1, [th1, th2]));
in
fun th1 COMP th2 = comp false th1 th2;
fun th1 INCR_COMP th2 = comp true (incr_indexes th2 th1) th2;
fun th1 COMP_INCR th2 = comp true th1 (incr_indexes th1 th2);
end;
fun comp_no_flatten (th, n) i rule =
(case distinct Thm.eq_thm (Seq.list_of
(Thm.bicompose {flatten = false, match = false, incremented = true}
(false, th, n) i (incr_indexes th rule))) of
[th'] => th'
| [] => raise THM ("comp_no_flatten", i, [th, rule])
| _ => raise THM ("comp_no_flatten: unique result expected", i, [th, rule]));
(** variations on Thm.instantiate **)
fun instantiate_normalize instpair th =
Thm.adjust_maxidx_thm ~1 (Thm.instantiate instpair th COMP_INCR asm_rl);
(*Left-to-right replacements: tpairs = [..., (vi, ti), ...].
Instantiates distinct Vars by terms, inferring type instantiations.*)
local
fun add_types (ct, cu) (thy, tye, maxidx) =
let
val {t, T, maxidx = maxt, ...} = Thm.rep_cterm ct;
val {t = u, T = U, maxidx = maxu, ...} = Thm.rep_cterm cu;
val maxi = Int.max (maxidx, Int.max (maxt, maxu));
val thy' = Theory.merge (thy, Theory.merge (Thm.theory_of_cterm ct, Thm.theory_of_cterm cu));
val (tye', maxi') = Sign.typ_unify thy' (T, U) (tye, maxi)
handle Type.TUNIFY => raise TYPE ("Ill-typed instantiation:\nType\n" ^
Syntax.string_of_typ_global thy' (Envir.norm_type tye T) ^
"\nof variable " ^
Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) t) ^
"\ncannot be unified with type\n" ^
Syntax.string_of_typ_global thy' (Envir.norm_type tye U) ^ "\nof term " ^
Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) u),
[T, U], [t, u])
in (thy', tye', maxi') end;
in
fun cterm_instantiate [] th = th
| cterm_instantiate ctpairs th =
let
val (thy, tye, _) = fold_rev add_types ctpairs (Thm.theory_of_thm th, Vartab.empty, 0);
val certT = ctyp_of thy;
val instT =
Vartab.fold (fn (xi, (S, T)) =>
cons (certT (TVar (xi, S)), certT (Envir.norm_type tye T))) tye [];
val inst = map (pairself (Thm.instantiate_cterm (instT, []))) ctpairs;
in instantiate_normalize (instT, inst) th end
handle TERM (msg, _) => raise THM (msg, 0, [th])
| TYPE (msg, _, _) => raise THM (msg, 0, [th]);
end;
(* instantiate by left-to-right occurrence of variables *)
fun instantiate' cTs cts thm =
let
fun err msg =
raise TYPE ("instantiate': " ^ msg,
map_filter (Option.map Thm.typ_of) cTs,
map_filter (Option.map Thm.term_of) cts);
fun inst_of (v, ct) =
(Thm.cterm_of (Thm.theory_of_cterm ct) (Var v), ct)
handle TYPE (msg, _, _) => err msg;
fun tyinst_of (v, cT) =
(Thm.ctyp_of (Thm.theory_of_ctyp cT) (TVar v), cT)
handle TYPE (msg, _, _) => err msg;
fun zip_vars xs ys =
zip_options xs ys handle ListPair.UnequalLengths =>
err "more instantiations than variables in thm";
(*instantiate types first!*)
val thm' =
if forall is_none cTs then thm
else Thm.instantiate
(map tyinst_of (zip_vars (rev (Thm.fold_terms Term.add_tvars thm [])) cTs), []) thm;
val thm'' =
if forall is_none cts then thm'
else Thm.instantiate
([], map inst_of (zip_vars (rev (Thm.fold_terms Term.add_vars thm' [])) cts)) thm';
in thm'' end;
(** renaming of bound variables **)
(* replace bound variables x_i in thm by y_i *)
(* where vs = [(x_1, y_1), ..., (x_n, y_n)] *)
fun rename_bvars [] thm = thm
| rename_bvars vs thm =
let
val cert = Thm.cterm_of (Thm.theory_of_thm thm);
fun ren (Abs (x, T, t)) = Abs (AList.lookup (op =) vs x |> the_default x, T, ren t)
| ren (t $ u) = ren t $ ren u
| ren t = t;
in Thm.equal_elim (Thm.reflexive (cert (ren (Thm.prop_of thm)))) thm end;
(* renaming in left-to-right order *)
fun rename_bvars' xs thm =
let
val cert = Thm.cterm_of (Thm.theory_of_thm thm);
val prop = Thm.prop_of thm;
fun rename [] t = ([], t)
| rename (x' :: xs) (Abs (x, T, t)) =
let val (xs', t') = rename xs t
in (xs', Abs (the_default x x', T, t')) end
| rename xs (t $ u) =
let
val (xs', t') = rename xs t;
val (xs'', u') = rename xs' u
in (xs'', t' $ u') end
| rename xs t = (xs, t);
in case rename xs prop of
([], prop') => Thm.equal_elim (Thm.reflexive (cert prop')) thm
| _ => error "More names than abstractions in theorem"
end;
end;
structure Basic_Drule: BASIC_DRULE = Drule;
open Basic_Drule;