--- a/Nominal/Ex/Typing.thy Tue Jan 04 13:47:38 2011 +0000
+++ b/Nominal/Ex/Typing.thy Wed Jan 05 16:51:27 2011 +0000
@@ -21,8 +21,11 @@
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})
@@ -48,8 +51,13 @@
|> 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]
+ if null avoid then [] else [vc_goal, finite_goal]
end
*}
@@ -136,6 +144,12 @@
*}
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} []
@@ -157,12 +171,16 @@
val prems' = prems
|> map (minus_permute_elim p)
|> map (eqvt_srule context)
-
+
val prm' = (prems' MRS prm)
|> flag ? (all_elims [p])
- |> flag ? (simplify (HOL_basic_ss addsimps @{thms permute_minus_cancel}))
+ |> flag ? (eqvt_srule context)
+
+ val _ = tracing ("prm':" ^ @{make_string} prm')
in
- simp_tac (HOL_ss addsimps (prm' :: @{thms induct_forall_def })) 1
+ 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
*}
@@ -178,7 +196,7 @@
val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
(* for inductive-premises*)
- fun tac1 prm = helper_tac true prm p context
+ fun tac1 prm = helper_tac true prm p context
(* for non-inductive premises *)
fun tac2 prm =
@@ -189,63 +207,120 @@
fun select prm (t, i) =
(if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
in
- EVERY1 [rtac prem', RANGE (map (SUBGOAL o select) prems) ]
+ EVERY1 [eqvt_stac ctxt, rtac prem', RANGE (map (SUBGOAL o select) prems) ]
end) ctxt
*}
ML {*
-fun fresh_thm ctxt fresh_thms p c prms avoid_trm =
+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) prms))) (Bound 0)
+ 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 _ = tracing ("fresh goal: " ^ Syntax.string_of_term ctxt fresh_goal)
+ 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})))
+ (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 binder_tac prem intr_cvars param_trms Ps fresh_thms avoid avoid_trm ctxt =
- Subgoal.FOCUS (fn {context, params, ...} =>
- let
- val thy = ProofContext.theory_of context
- 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 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
- val avoid_trm' = subst_free (param_trms ~~ prm_trms) avoid_trm
-
- val fthm = fresh_thm context fresh_thms (term_of p) (term_of c) (map term_of prms) avoid_trm'
- in
- Skip_Proof.cheat_tac thy
- (* EVERY1 [rtac prem'] *)
- end) ctxt
-*}
-
-ML {*
-fun case_tac ctxt fresh_thms Ps (avoid, avoid_trm) intr_cvars param_trms prem =
+fun prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms concl_args
+ {prems, context} =
let
- val tac1 = non_binder_tac prem intr_cvars Ps ctxt
- val tac2 = binder_tac prem intr_cvars param_trms Ps fresh_thms avoid avoid_trm ctxt
- in
- EVERY' [ rtac @{thm allI}, rtac @{thm allI}, eqvt_stac ctxt,
- if null avoid then tac1 else tac2 ]
- end
-*}
-
-ML {*
-fun prove_sinduct_tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars param_trms {prems, context} =
- let
- val cases_tac = map4 (case_tac context fresh_thms Ps) (avoids ~~avoid_trms) intr_cvars param_trms prems
+ 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
@@ -255,8 +330,10 @@
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 rule_names avoids raw_induct intrs ctxt =
+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
@@ -311,10 +388,10 @@
val ind_prems' = ind_prems
|> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)
- fun after_qed ctxt_outside fresh_thms ctxt =
+ fun after_qed ctxt_outside user_thms ctxt =
let
- val thms = Goal.prove ctxt [] ind_prems' ind_concl'
- (prove_sinduct_tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars param_trms)
+ 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)
@@ -322,10 +399,22 @@
|> map (Drule.rotate_prems (length ind_prems'))
|> map zero_var_indexes
- val _ = tracing ("RESULTS\n" ^ cat_lines (map (Syntax.string_of_term ctxt o prop_of) thms))
+ 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
- end
+ |> 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
@@ -366,7 +455,7 @@
val avoid_trms = map2 read_avoids avoids_ordered intrs
in
- prove_strong_inductive rule_names avoid_trms raw_induct intrs ctxt
+ prove_strong_inductive names rule_names avoid_trms raw_induct intrs ctxt
end
*}
@@ -422,6 +511,8 @@
nominal_inductive Acc .
+thm Acc.strong_induct
+
section {* Typing *}
nominal_datatype ty =
@@ -441,8 +532,8 @@
inductive
valid :: "(name \<times> ty) list \<Rightarrow> bool"
where
- "valid []"
-| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
+ v_Nil[intro]: "valid []"
+| v_Cons[intro]: "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
inductive
typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
@@ -459,105 +550,66 @@
equivariance valid
equivariance typing
-
nominal_inductive typing
avoids t_Lam: "x"
- (* | t_Var: "x" *)
apply -
apply(simp_all add: fresh_star_def ty_fresh lam.fresh)?
done
+thm typing.strong_induct
+abbreviation
+ "sub_context" :: "(name \<times> ty) list \<Rightarrow> (name \<times> ty) list \<Rightarrow> bool" ("_ \<subseteq> _" [60,60] 60)
+where
+ "\<Gamma>1 \<subseteq> \<Gamma>2 \<equiv> \<forall>x T. (x, T) \<in> set \<Gamma>1 \<longrightarrow> (x, T) \<in> set \<Gamma>2"
+
+text {* Now it comes: The Weakening Lemma *}
+
+text {*
+ The first version is, after setting up the induction,
+ completely automatic except for use of atomize. *}
-lemma
- fixes c::"'a::fs"
- assumes a: "\<Gamma> \<turnstile> t : T"
- and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
- and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk>
- \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
- and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>{atom x} \<sharp>* c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk>
- \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
- shows "P c \<Gamma> t T"
-proof -
- from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
- proof (induct)
- case (t_Var \<Gamma> x T p c)
- then show ?case
- apply -
- apply(perm_strict_simp)
- thm a1
- apply(rule a1)
- apply(drule_tac p="p" in permute_boolI)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- apply(rotate_tac 1)
- apply(drule_tac p="p" in permute_boolI)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- done
- next
- case (t_App \<Gamma> t1 T1 T2 t2 p c)
- then show ?case
- apply -
- apply(perm_strict_simp)
- apply(rule a2)
- apply(drule_tac p="p" in permute_boolI)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- apply(assumption)
- apply(rotate_tac 2)
- apply(drule_tac p="p" in permute_boolI)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- apply(assumption)
- done
- next
- case (t_Lam x \<Gamma> T1 t T2 p c)
- then show ?case
- apply -
- apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and>
- supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
- apply(erule exE)
- apply(rule_tac t="p \<bullet> \<Gamma>" and s="(q + p) \<bullet> \<Gamma>" in subst)
- apply(simp only: permute_plus)
- apply(rule supp_perm_eq)
- apply(simp add: supp_Pair fresh_star_Un)
- apply(rule_tac t="p \<bullet> Lam x t" and s="(q + p) \<bullet> Lam x t" in subst)
- apply(simp only: permute_plus)
- apply(rule supp_perm_eq)
- apply(simp add: supp_Pair fresh_star_Un)
- apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
- apply(simp only: permute_plus)
- apply(rule supp_perm_eq)
- apply(simp add: supp_Pair fresh_star_Un)
- (* apply(perm_simp) *)
- apply(simp (no_asm) only: eqvts)
- apply(rule a3)
- apply(simp only: eqvts permute_plus)
- apply(rule_tac p="- (q + p)" in permute_boolE)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- apply(rule_tac p="- (q + p)" in permute_boolE)
- apply(perm_strict_simp add: permute_minus_cancel)
- apply(assumption)
- apply(perm_strict_simp)
- apply(simp only:)
- thm at_set_avoiding2
- apply(rule at_set_avoiding2)
- apply(simp add: finite_supp)
- apply(simp add: finite_supp)
- apply(simp add: finite_supp)
- apply(rule_tac p="-p" in permute_boolE)
- apply(perm_strict_simp add: permute_minus_cancel)
- --"supplied by the user"
- apply(simp add: fresh_star_Pair)
- sorry
- qed
- then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
- then show "P c \<Gamma> t T" by simp
-qed
+lemma weakening_version2:
+ fixes \<Gamma>1 \<Gamma>2::"(name \<times> ty) list"
+ and t ::"lam"
+ and \<tau> ::"ty"
+ assumes a: "\<Gamma>1 \<turnstile> t : T"
+ and b: "valid \<Gamma>2"
+ and c: "\<Gamma>1 \<subseteq> \<Gamma>2"
+ shows "\<Gamma>2 \<turnstile> t : T"
+using a b c
+proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
+ case (t_Var \<Gamma>1 x T) (* variable case *)
+ have "\<Gamma>1 \<subseteq> \<Gamma>2" by fact
+ moreover
+ have "valid \<Gamma>2" by fact
+ moreover
+ have "(x,T)\<in> set \<Gamma>1" by fact
+ ultimately show "\<Gamma>2 \<turnstile> Var x : T" by auto
+next
+ case (t_Lam x \<Gamma>1 T1 t T2) (* lambda case *)
+ have vc: "atom x \<sharp> \<Gamma>2" by fact (* variable convention *)
+ have ih: "\<lbrakk>valid ((x, T1) # \<Gamma>2); (x, T1) # \<Gamma>1 \<subseteq> (x, T1) # \<Gamma>2\<rbrakk> \<Longrightarrow> (x, T1) # \<Gamma>2 \<turnstile> t : T2" by fact
+ have "\<Gamma>1 \<subseteq> \<Gamma>2" by fact
+ then have "(x, T1) # \<Gamma>1 \<subseteq> (x, T1) # \<Gamma>2" by simp
+ moreover
+ have "valid \<Gamma>2" by fact
+ then have "valid ((x, T1) # \<Gamma>2)" using vc by (simp add: v_Cons)
+ ultimately have "(x, T1) # \<Gamma>2 \<turnstile> t : T2" using ih by simp
+ with vc show "\<Gamma>2 \<turnstile> Lam x t : T1 \<rightarrow> T2" by auto
+qed (auto) (* app case *)
-*)
+lemma weakening_version1:
+ fixes \<Gamma>1 \<Gamma>2::"(name \<times> ty) list"
+ assumes a: "\<Gamma>1 \<turnstile> t : T"
+ and b: "valid \<Gamma>2"
+ and c: "\<Gamma>1 \<subseteq> \<Gamma>2"
+ shows "\<Gamma>2 \<turnstile> t : T"
+using a b c
+apply (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
+apply (auto | atomize)+
+done
+
end