--- a/Nominal/Ex/Typing.thy Wed Jan 05 16:51:27 2011 +0000
+++ b/Nominal/Ex/Typing.thy Wed Jan 05 17:33:43 2011 +0000
@@ -20,498 +20,6 @@
thm lam.fv_bn_eqvt
thm lam.size_eqvt
-ML {*
-fun mk_cplus p q = Thm.capply (Thm.capply @{cterm "plus::perm \<Rightarrow> perm \<Rightarrow> perm"} p) q
-
-fun mk_cminus p = Thm.capply @{cterm "uminus::perm \<Rightarrow> perm"} p
-
-
-fun minus_permute_intro_tac p =
- rtac (Drule.instantiate' [] [SOME (mk_cminus p)] @{thm permute_boolE})
-
-fun minus_permute_elim p thm =
- thm RS (Drule.instantiate' [] [NONE, SOME (mk_cminus p)] @{thm permute_boolI})
-*}
-
-ML {*
-fun real_head_of (@{term Trueprop} $ t) = real_head_of t
- | real_head_of (Const ("==>", _) $ _ $ t) = real_head_of t
- | real_head_of (Const (@{const_name all}, _) $ Abs (_, _, t)) = real_head_of t
- | real_head_of (Const (@{const_name All}, _) $ Abs (_, _, t)) = real_head_of t
- | real_head_of (Const ("HOL.induct_forall", _) $ Abs (_, _, t)) = real_head_of t
- | real_head_of t = head_of t
-*}
-
-ML {*
-fun mk_vc_compat (avoid, avoid_trm) prems concl_args params =
- let
- val vc_goal = concl_args
- |> HOLogic.mk_tuple
- |> mk_fresh_star avoid_trm
- |> HOLogic.mk_Trueprop
- |> (curry Logic.list_implies) prems
- |> (curry list_all_free) params
- val finite_goal = avoid_trm
- |> mk_finite
- |> HOLogic.mk_Trueprop
- |> (curry Logic.list_implies) prems
- |> (curry list_all_free) params
- in
- if null avoid then [] else [vc_goal, finite_goal]
- end
-*}
-
-ML {*
-fun map_term prop f trm =
- if prop trm
- then f trm
- else case trm of
- (t1 $ t2) => map_term prop f t1 $ map_term prop f t2
- | Abs (x, T, t) => Abs (x, T, map_term prop f t)
- | _ => trm
-*}
-
-ML {*
-fun add_p_c p (c, c_ty) trm =
- let
- val (P, args) = strip_comb trm
- val (P_name, P_ty) = dest_Free P
- val (ty_args, bool) = strip_type P_ty
- val args' = map (mk_perm p) args
- in
- list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')
- |> (fn t => HOLogic.all_const c_ty $ lambda c t )
- |> (fn t => HOLogic.all_const @{typ perm} $ lambda p t)
- end
-*}
-
-ML {*
-fun induct_forall_const T = Const ("HOL.induct_forall", (T --> @{typ bool}) --> @{typ bool})
-fun mk_induct_forall (a, T) t = induct_forall_const T $ Abs (a, T, t)
-*}
-
-ML {*
-fun add_c_prop qnt Ps (c, c_name, c_ty) trm =
- let
- fun add t =
- let
- val (P, args) = strip_comb t
- val (P_name, P_ty) = dest_Free P
- val (ty_args, bool) = strip_type P_ty
- val args' = args
- |> qnt ? map (incr_boundvars 1)
- in
- list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')
- |> qnt ? mk_induct_forall (c_name, c_ty)
- end
- in
- map_term (member (op =) Ps o head_of) add trm
- end
-*}
-
-ML {*
-fun prep_prem Ps c_name c_ty (avoid, avoid_trm) (params, prems, concl) =
- let
- val prems' = prems
- |> map (incr_boundvars 1)
- |> map (add_c_prop true Ps (Bound 0, c_name, c_ty))
-
- val avoid_trm' = avoid_trm
- |> (curry list_abs_free) (params @ [(c_name, c_ty)])
- |> strip_abs_body
- |> (fn t => mk_fresh_star_ty c_ty t (Bound 0))
- |> HOLogic.mk_Trueprop
-
- val prems'' =
- if null avoid
- then prems'
- else avoid_trm' :: prems'
-
- val concl' = concl
- |> incr_boundvars 1
- |> add_c_prop false Ps (Bound 0, c_name, c_ty)
- in
- mk_full_horn (params @ [(c_name, c_ty)]) prems'' concl'
- end
-*}
-
-
-ML {*
-fun same_name (Free (a1, _), Free (a2, _)) = (a1 = a2)
- | same_name (Var (a1, _), Var (a2, _)) = (a1 = a2)
- | same_name (Const (a1, _), Const (a2, _)) = (a1 = a2)
- | same_name _ = false
-*}
-
-ML {*
-fun map7 _ [] [] [] [] [] [] [] = []
- | map7 f (x :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) (r :: rs) (s :: ss) =
- f x y z u v r s :: map7 f xs ys zs us vs rs ss
-*}
-
-ML {*
-(* local abbreviations *)
-fun eqvt_stac ctxt = Nominal_Permeq.eqvt_strict_tac ctxt @{thms permute_minus_cancel} []
-fun eqvt_srule ctxt = Nominal_Permeq.eqvt_strict_rule ctxt @{thms permute_minus_cancel} []
-*}
-
-ML {*
-val all_elims =
- let
- fun spec' ct = Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec}
- in
- fold (fn ct => fn th => th RS spec' ct)
- end
-*}
-
-ML {*
-fun helper_tac flag prm p ctxt =
- Subgoal.SUBPROOF (fn {context, prems, ...} =>
- let
- val prems' = prems
- |> map (minus_permute_elim p)
- |> map (eqvt_srule context)
-
- val prm' = (prems' MRS prm)
- |> flag ? (all_elims [p])
- |> flag ? (eqvt_srule context)
-
- val _ = tracing ("prm':" ^ @{make_string} prm')
- in
- print_tac "start helper"
- THEN asm_full_simp_tac (HOL_ss addsimps (prm' :: @{thms induct_forall_def})) 1
- THEN print_tac "final helper"
- end) ctxt
-*}
-
-ML {*
-fun non_binder_tac prem intr_cvars Ps ctxt =
- Subgoal.SUBPROOF (fn {context, params, prems, ...} =>
- let
- val thy = ProofContext.theory_of context
- val (prms, p, _) = split_last2 (map snd params)
- val prm_tys = map (fastype_of o term_of) prms
- val cperms = map (cterm_of thy o perm_const) prm_tys
- val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms
- val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
-
- (* for inductive-premises*)
- fun tac1 prm = helper_tac true prm p context
-
- (* for non-inductive premises *)
- fun tac2 prm =
- EVERY' [ minus_permute_intro_tac p,
- eqvt_stac context,
- helper_tac false prm p context ]
-
- fun select prm (t, i) =
- (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
- in
- EVERY1 [eqvt_stac ctxt, rtac prem', RANGE (map (SUBGOAL o select) prems) ]
- end) ctxt
-*}
-
-
-ML {*
-fun fresh_thm ctxt user_thm p c concl_args avoid_trm =
- let
- val conj1 =
- mk_fresh_star (mk_perm (Bound 0) (mk_perm p avoid_trm)) c
- val conj2 =
- mk_fresh_star_ty @{typ perm} (mk_supp (HOLogic.mk_tuple (map (mk_perm p) concl_args))) (Bound 0)
- val fresh_goal = mk_exists ("q", @{typ perm}) (HOLogic.mk_conj (conj1, conj2))
- |> HOLogic.mk_Trueprop
-
- val ss = @{thms finite_supp supp_Pair finite_Un permute_finite} @
- @{thms fresh_star_Pair fresh_star_permute_iff}
- val simp = asm_full_simp_tac (HOL_ss addsimps ss)
- in
- Goal.prove ctxt [] [] fresh_goal
- (K (HEADGOAL (rtac @{thm at_set_avoiding2}
- THEN_ALL_NEW EVERY' [cut_facts_tac user_thm, REPEAT o etac @{thm conjE}, simp])))
- end
-*}
-
-ML {*
-val supp_perm_eq' =
- @{lemma "supp (p \<bullet> x) \<sharp>* q ==> p \<bullet> x == (q + p) \<bullet> x" by (simp add: supp_perm_eq)}
-val fresh_star_plus =
- @{lemma "(q \<bullet> (p \<bullet> x)) \<sharp>* c ==> ((q + p) \<bullet> x) \<sharp>* c" by (simp add: permute_plus)}
-*}
-
-ML {*
-fun binder_tac prem intr_cvars param_trms Ps user_thm avoid avoid_trm concl_args ctxt =
- Subgoal.FOCUS (fn {context = ctxt, params, prems, concl, ...} =>
- let
- val thy = ProofContext.theory_of ctxt
- val (prms, p, c) = split_last2 (map snd params)
- val prm_trms = map term_of prms
- val prm_tys = map fastype_of prm_trms
-
- val avoid_trm' = subst_free (param_trms ~~ prm_trms) avoid_trm
- val concl_args' = map (subst_free (param_trms ~~ prm_trms)) concl_args
-
- val user_thm' = map (cterm_instantiate (intr_cvars ~~ prms)) user_thm
- |> map (full_simplify (HOL_ss addsimps (@{thm fresh_star_Pair}::prems)))
-
- val fthm = fresh_thm ctxt user_thm' (term_of p) (term_of c) concl_args' avoid_trm'
-
- val (([(_, q)], fprop :: fresh_eqs), ctxt') = Obtain.result
- (K (EVERY1 [etac @{thm exE},
- full_simp_tac (HOL_basic_ss addsimps @{thms supp_Pair fresh_star_Un}),
- REPEAT o etac @{thm conjE},
- dtac fresh_star_plus,
- REPEAT o dtac supp_perm_eq'])) [fthm] ctxt
-
- val expand_conv = Conv.try_conv (Conv.rewrs_conv fresh_eqs)
- fun expand_conv_bot ctxt = Conv.bottom_conv (K expand_conv) ctxt
-
- val cperms = map (cterm_of thy o perm_const) prm_tys
- val qp_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 (mk_cplus q p) ct2) cperms prms
- val prem' = cterm_instantiate (intr_cvars ~~ qp_prms) prem
-
- val fprop' = eqvt_srule ctxt' fprop
- val tac_fresh = simp_tac (HOL_basic_ss addsimps [fprop'])
-
- (* for inductive-premises*)
- fun tac1 prm = helper_tac true prm (mk_cplus q p) ctxt'
-
- (* for non-inductive premises *)
- fun tac2 prm =
- EVERY' [ minus_permute_intro_tac (mk_cplus q p),
- eqvt_stac ctxt,
- helper_tac false prm (mk_cplus q p) ctxt' ]
-
- fun select prm (t, i) =
- (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
-
- val _ = tracing ("fthm:\n" ^ @{make_string} fthm)
- val _ = tracing ("fr_eqs:\n" ^ cat_lines (map @{make_string} fresh_eqs))
- val _ = tracing ("fprop:\n" ^ @{make_string} fprop)
- val _ = tracing ("fprop':\n" ^ @{make_string} fprop')
- val _ = tracing ("fperm:\n" ^ @{make_string} q)
- val _ = tracing ("prem':\n" ^ @{make_string} prem')
-
- val side_thm = Goal.prove ctxt' [] [] (term_of concl)
- (fn {context, ...} =>
- EVERY1 [ CONVERSION (expand_conv_bot context),
- eqvt_stac context,
- rtac prem',
- RANGE (tac_fresh :: map (SUBGOAL o select) prems),
- K (print_tac "GOAL") ])
- |> singleton (ProofContext.export ctxt' ctxt)
- in
- rtac side_thm 1
- end) ctxt
-*}
-
-ML {*
-fun case_tac ctxt Ps avoid avoid_trm intr_cvars param_trms prem user_thm concl_args =
- let
- val tac1 = non_binder_tac prem intr_cvars Ps ctxt
- val tac2 = binder_tac prem intr_cvars param_trms Ps user_thm avoid avoid_trm concl_args ctxt
- in
- EVERY' [ rtac @{thm allI}, rtac @{thm allI}, if null avoid then tac1 else tac2 ]
- end
-*}
-
-ML {*
-fun prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms concl_args
- {prems, context} =
- let
- val cases_tac =
- map7 (case_tac context Ps) avoids avoid_trms intr_cvars param_trms prems user_thms concl_args
- in
- EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]
- end
-*}
-
-ML {*
-val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (!!c. Q ==> P (0::perm) c)" by simp}
-*}
-
-ML {* Local_Theory.note *}
-
-ML {*
-fun prove_strong_inductive pred_names rule_names avoids raw_induct intrs ctxt =
- let
- val thy = ProofContext.theory_of ctxt
- val ((_, [raw_induct']), ctxt') = Variable.import true [raw_induct] ctxt
-
- val (ind_prems, ind_concl) = raw_induct'
- |> prop_of
- |> Logic.strip_horn
- |>> map strip_full_horn
- val params = map (fn (x, _, _) => x) ind_prems
- val param_trms = (map o map) Free params
-
- val intr_vars_tys = map (fn t => rev (Term.add_vars (prop_of t) [])) intrs
- val intr_vars = (map o map) fst intr_vars_tys
- val intr_vars_substs = map2 (curry (op ~~)) intr_vars param_trms
- val intr_cvars = (map o map) (cterm_of thy o Var) intr_vars_tys
-
- val (intr_prems, intr_concls) = intrs
- |> map prop_of
- |> map2 subst_Vars intr_vars_substs
- |> map Logic.strip_horn
- |> split_list
-
- val intr_concls_args = map (snd o strip_comb o HOLogic.dest_Trueprop) intr_concls
-
- val avoid_trms = avoids
- |> (map o map) (setify ctxt')
- |> map fold_union
-
- val vc_compat_goals =
- map4 mk_vc_compat (avoids ~~ avoid_trms) intr_prems intr_concls_args params
-
- val ([c_name, a, p], ctxt'') = Variable.variant_fixes ["c", "'a", "p"] ctxt'
- val c_ty = TFree (a, @{sort fs})
- val c = Free (c_name, c_ty)
- val p = Free (p, @{typ perm})
-
- val (preconds, ind_concls) = ind_concl
- |> HOLogic.dest_Trueprop
- |> HOLogic.dest_conj
- |> map HOLogic.dest_imp
- |> split_list
-
- val Ps = map (fst o strip_comb) ind_concls
-
- val ind_concl' = ind_concls
- |> map (add_p_c p (c, c_ty))
- |> (curry (op ~~)) preconds
- |> map HOLogic.mk_imp
- |> fold_conj
- |> HOLogic.mk_Trueprop
-
- val ind_prems' = ind_prems
- |> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)
-
- fun after_qed ctxt_outside user_thms ctxt =
- let
- val strong_ind_thms = Goal.prove ctxt [] ind_prems' ind_concl'
- (prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms intr_concls_args)
- |> singleton (ProofContext.export ctxt ctxt_outside)
- |> Datatype_Aux.split_conj_thm
- |> map (fn thm => thm RS normalise)
- |> map (asm_full_simplify (HOL_basic_ss addsimps @{thms permute_zero induct_rulify}))
- |> map (Drule.rotate_prems (length ind_prems'))
- |> map zero_var_indexes
-
- val qualified_thm_name = pred_names
- |> map Long_Name.base_name
- |> space_implode "_"
- |> (fn s => Binding.qualify false s (Binding.name "strong_induct"))
-
- val attrs =
- [ Attrib.internal (K (Rule_Cases.consumes 1)),
- Attrib.internal (K (Rule_Cases.case_names rule_names)) ]
- val _ = tracing ("RESULTS\n" ^ cat_lines (map (Syntax.string_of_term ctxt o prop_of) strong_ind_thms))
- val _ = tracing ("rule_names: " ^ commas rule_names)
- val _ = tracing ("pred_names: " ^ commas pred_names)
- in
- ctxt
- |> Local_Theory.note ((qualified_thm_name, attrs), strong_ind_thms)
- |> snd
- end
- in
- Proof.theorem NONE (after_qed ctxt) ((map o map) (rpair []) vc_compat_goals) ctxt''
- end
-*}
-
-ML {*
-fun prove_strong_inductive_cmd (pred_name, avoids) ctxt =
- let
- val thy = ProofContext.theory_of ctxt;
- val ({names, ...}, {raw_induct, intrs, ...}) =
- Inductive.the_inductive ctxt (Sign.intern_const thy pred_name);
-
- val rule_names =
- hd names
- |> the o Induct.lookup_inductP ctxt
- |> fst o Rule_Cases.get
- |> map fst
-
- val _ = (case duplicates (op = o pairself fst) avoids of
- [] => ()
- | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)))
-
- val _ = (case subtract (op =) rule_names (map fst avoids) of
- [] => ()
- | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs))
-
- val avoids_ordered = order_default (op =) [] rule_names avoids
-
- fun read_avoids avoid_trms intr =
- let
- (* fixme hack *)
- val (((_, ctrms), _), ctxt') = Variable.import true [intr] ctxt
- val trms = map (term_of o snd) ctrms
- val ctxt'' = fold Variable.declare_term trms ctxt'
- in
- map (Syntax.read_term ctxt'') avoid_trms
- end
-
- val avoid_trms = map2 read_avoids avoids_ordered intrs
- in
- prove_strong_inductive names rule_names avoid_trms raw_induct intrs ctxt
- end
-*}
-
-ML {*
-(* outer syntax *)
-local
- structure P = Parse;
- structure S = Scan
-
- val _ = Keyword.keyword "avoids"
-
- val single_avoid_parser =
- P.name -- (P.$$$ ":" |-- P.and_list1 P.term)
-
- val avoids_parser =
- S.optional (P.$$$ "avoids" |-- P.enum1 "|" single_avoid_parser) []
-
- val main_parser = P.xname -- avoids_parser
-in
- val _ =
- Outer_Syntax.local_theory_to_proof "nominal_inductive"
- "prove strong induction theorem for inductive predicate involving nominal datatypes"
- Keyword.thy_goal (main_parser >> prove_strong_inductive_cmd)
-end
-*}
-
-inductive
- Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
-where
- AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"
-
-(*
-equivariance Acc
-*)
-
-lemma Acc_eqvt [eqvt]:
- fixes p::"perm"
- assumes a: "Acc R x"
- shows "Acc (p \<bullet> R) (p \<bullet> x)"
-using a
-apply(induct)
-apply(rule AccI)
-apply(rotate_tac 1)
-apply(drule_tac x="-p \<bullet> y" in meta_spec)
-apply(simp)
-apply(drule meta_mp)
-apply(rule_tac p="p" in permute_boolE)
-apply(perm_simp add: permute_minus_cancel)
-apply(assumption)
-apply(assumption)
-done
-
-
-nominal_inductive Acc .
-
-thm Acc.strong_induct
section {* Typing *}
@@ -527,8 +35,6 @@
apply (simp_all add: ty.fresh pure_fresh)
done
-
-
inductive
valid :: "(name \<times> ty) list \<Rightarrow> bool"
where
@@ -545,16 +51,13 @@
thm typing.intros
thm typing.induct
-
-
equivariance valid
equivariance typing
nominal_inductive typing
avoids t_Lam: "x"
- apply -
- apply(simp_all add: fresh_star_def ty_fresh lam.fresh)?
- done
+ by (simp_all add: fresh_star_def ty_fresh lam.fresh)
+
thm typing.strong_induct
@@ -611,6 +114,34 @@
done
+inductive
+ Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+where
+ AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"
+
+
+lemma Acc_eqvt [eqvt]:
+ fixes p::"perm"
+ assumes a: "Acc R x"
+ shows "Acc (p \<bullet> R) (p \<bullet> x)"
+using a
+apply(induct)
+apply(rule AccI)
+apply(rotate_tac 1)
+apply(drule_tac x="-p \<bullet> y" in meta_spec)
+apply(simp)
+apply(drule meta_mp)
+apply(rule_tac p="p" in permute_boolE)
+apply(perm_simp add: permute_minus_cancel)
+apply(assumption)
+apply(assumption)
+done
+
+
+nominal_inductive Acc .
+
+thm Acc.strong_induct
+
end