--- a/Nominal/Ex/Lambda.thy Mon Jan 17 12:34:11 2011 +0000
+++ b/Nominal/Ex/Lambda.thy Mon Jan 17 12:37:37 2011 +0000
@@ -2,11 +2,8 @@
imports "../Nominal2"
begin
-
atom_decl name
-ML {* suffix *}
-
nominal_datatype lam =
Var "name"
| App "lam" "lam"
@@ -22,460 +19,7 @@
thm lam.fv_bn_eqvt
thm lam.size_eqvt
-definition
- "eqvt_at f x \<equiv> \<forall>p. p \<bullet> (f x) = f (p \<bullet> x)"
-function
- depth :: "lam \<Rightarrow> nat"
-where
- "depth (Var x) = 1"
-| "depth (App t1 t2) = (max (depth t1) (depth t2)) + 1"
-| "depth (Lam x t) = (depth t) + 1"
-oops
-
-section {* Matching *}
-
-definition
- MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"
-where
- "MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"
-
-(*
-lemma MATCH_eqvt:
- shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"
-unfolding MATCH_def
-apply(perm_simp the_eqvt)
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-apply(simp)
-thm eqvts_raw
-apply(subst if_eqvt)
-apply(subst ex1_eqvt)
-apply(subst permute_fun_def)
-apply(subst ex_eqvt)
-apply(subst permute_fun_def)
-apply(subst eq_eqvt)
-apply(subst permute_fun_app_eq[where f="M"])
-apply(simp only: permute_minus_cancel)
-apply(subst permute_prod.simps)
-apply(subst permute_prod.simps)
-apply(simp only: permute_minus_cancel)
-apply(simp only: permute_bool_def)
-apply(simp)
-apply(subst ex1_eqvt)
-apply(subst permute_fun_def)
-apply(subst ex_eqvt)
-apply(subst permute_fun_def)
-apply(subst eq_eqvt)
-
-apply(simp only: eqvts)
-apply(simp)
-apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")
-apply(drule sym)
-apply(simp)
-apply(rule impI)
-apply(simp add: perm_bool)
-apply(rule trans)
-apply(rule pt_the_eqvt[OF pta at])
-apply(assumption)
-apply(simp add: pt_ex_eqvt[OF pt at])
-apply(simp add: pt_eq_eqvt[OF ptb at])
-apply(rule cheat)
-apply(rule trans)
-apply(rule pt_ex1_eqvt)
-apply(rule pta)
-apply(rule at)
-apply(simp add: pt_ex_eqvt[OF pt at])
-apply(simp add: pt_eq_eqvt[OF ptb at])
-apply(subst pt_pi_rev[OF pta at])
-apply(subst pt_fun_app_eq[OF pt at])
-apply(subst pt_pi_rev[OF pt at])
-apply(simp)
-done
-
-lemma MATCH_cng:
- assumes a: "M1 = M2" "d1 = d2"
- shows "MATCH M1 d1 x = MATCH M2 d2 x"
-using a by simp
-
-lemma MATCH_eq:
- assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"
- shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"
-using a
-unfolding MATCH_def
-apply(subst if_P)
-apply(rule_tac a="r x" in ex1I)
-apply(rule_tac x="x" in exI)
-apply(blast)
-apply(erule exE)
-apply(drule_tac x="q" in meta_spec)
-apply(auto)[1]
-apply(rule the_equality)
-apply(blast)
-apply(erule exE)
-apply(drule_tac x="q" in meta_spec)
-apply(auto)[1]
-done
-
-lemma MATCH_eq2:
- assumes a: "t = l x1 x2" "G x1 x2" "\<And>x1' x2'. t = l x1' x2' \<Longrightarrow> G x1' x2' \<Longrightarrow> r x1' x2' = r x1 x2"
- shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = r x1 x2"
-sorry
-
-lemma MATCH_neq:
- assumes a: "\<And>x. t = l x \<Longrightarrow> G x \<Longrightarrow> False"
- shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = d"
-using a
-unfolding MATCH_def
-apply(subst if_not_P)
-apply(blast)
-apply(rule refl)
-done
-
-lemma MATCH_neq2:
- assumes a: "\<And>x1 x2. t = l x1 x2 \<Longrightarrow> G x1 x2 \<Longrightarrow> False"
- shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = d"
-using a
-unfolding MATCH_def
-apply(subst if_not_P)
-apply(auto)
-done
-*)
-
-ML {*
-fun mk_avoids ctxt params name set =
- let
- val (_, ctxt') = ProofContext.add_fixes
- (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
- fun mk s =
- let
- val t = Syntax.read_term ctxt' s;
- val t' = list_abs_free (params, t) |>
- funpow (length params) (fn Abs (_, _, t) => t)
- in (t', HOLogic.dest_setT (fastype_of t)) end
- handle TERM _ =>
- error ("Expression " ^ quote s ^ " to be avoided in case " ^
- quote name ^ " is not a set type");
- fun add_set p [] = [p]
- | add_set (t, T) ((u, U) :: ps) =
- if T = U then
- let val S = HOLogic.mk_setT T
- in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
- end
- else (u, U) :: add_set (t, T) ps
- in
- (mk #> add_set) set
- end;
-*}
-
-
-ML {*
- writeln (commas (map (Syntax.string_of_term @{context} o fst)
- (mk_avoids @{context} [] "t_Var" "{x}" [])))
-*}
-
-
-ML {*
-
-fun prove_strong_ind (pred_name, avoids) ctxt =
- Proof.theorem NONE (K I) [] ctxt
-
-local structure P = Parse and K = Keyword in
-
-val _ =
- Outer_Syntax.local_theory_to_proof "nominal_inductive"
- "proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
- (P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
- (P.$$$ ":" |-- P.and_list1 P.term))) []) >> prove_strong_ind)
-
-end;
-
-*}
-
-(*
-nominal_inductive typing
-*)
-
-(* Substitution *)
-
-primrec match_Var_raw where
- "match_Var_raw (Var_raw x) = Some x"
-| "match_Var_raw (App_raw x y) = None"
-| "match_Var_raw (Lam_raw n t) = None"
-
-quotient_definition
- "match_Var :: lam \<Rightarrow> name option"
-is match_Var_raw
-
-lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
- apply rule
- apply (induct_tac x y rule: alpha_lam_raw.induct)
- apply simp_all
- done
-
-lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
-
-primrec match_App_raw where
- "match_App_raw (Var_raw x) = None"
-| "match_App_raw (App_raw x y) = Some (x, y)"
-| "match_App_raw (Lam_raw n t) = None"
-
-(*
-quotient_definition
- "match_App :: lam \<Rightarrow> (lam \<times> lam) option"
-is match_App_raw
-
-lemma [quot_respect]:
- "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
- apply (intro fun_relI)
- apply (induct_tac a b rule: alpha_lam_raw.induct)
- apply simp_all
- done
-
-lemmas match_App_simps = match_App_raw.simps[quot_lifted]
-
-definition new where
- "new (s :: 'a :: fs) = (THE x. \<forall>a \<in> supp s. atom x \<noteq> a)"
-
-definition
- "match_Lam (S :: 'a :: fs) t = (if (\<exists>n s. (t = Lam n s)) then
- (let z = new (S, t) in Some (z, THE s. t = Lam z s)) else None)"
-
-lemma lam_half_inj: "(Lam z s = Lam z sa) = (s = sa)"
- apply auto
- apply (simp only: lam.eq_iff alphas)
- apply clarify
- apply (simp add: eqvts)
- sorry
-
-lemma match_Lam_simps:
- "match_Lam S (Var n) = None"
- "match_Lam S (App l r) = None"
- "z = new (S, (Lam z s)) \<Longrightarrow> match_Lam S (Lam z s) = Some (z, s)"
- apply (simp_all add: match_Lam_def)
- apply (simp add: lam_half_inj)
- apply auto
- done
-*)
-(*
-lemma match_Lam_simps2:
- "atom n \<sharp> ((S :: 'a :: fs), Lam n s) \<Longrightarrow> match_Lam S (Lam n s) = Some (n, s)"
- apply (rule_tac t="Lam n s"
- and s="Lam (new (S, (Lam n s))) ((n \<leftrightarrow> (new (S, (Lam n s)))) \<bullet> s)" in subst)
- defer
- apply (subst match_Lam_simps(3))
- defer
- apply simp
-*)
-
-(*primrec match_Lam_raw where
- "match_Lam_raw (S :: atom set) (Var_raw x) = None"
-| "match_Lam_raw S (App_raw x y) = None"
-| "match_Lam_raw S (Lam_raw n t) = (let z = new (S \<union> (fv_lam_raw t - {atom n})) in Some (z, (n \<leftrightarrow> z) \<bullet> t))"
-
-quotient_definition
- "match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
-is match_Lam_raw
-
-lemma swap_fresh:
- assumes a: "fv_lam_raw t \<sharp>* p"
- shows "alpha_lam_raw (p \<bullet> t) t"
- using a apply (induct t)
- apply (simp add: supp_at_base fresh_star_def)
- apply (rule alpha_lam_raw.intros)
- apply (metis Rep_name_inverse atom_eqvt atom_name_def fresh_perm)
- apply (simp)
- apply (simp only: fresh_star_union)
- apply clarify
- apply (rule alpha_lam_raw.intros)
- apply simp
- apply simp
- apply simp
- apply (rule alpha_lam_raw.intros)
- sorry
-
-lemma [quot_respect]:
- "(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
- proof (intro fun_relI, clarify)
- fix S t s
- assume a: "alpha_lam_raw t s"
- show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
- using a proof (induct t s rule: alpha_lam_raw.induct)
- case goal1 show ?case by simp
- next
- case goal2 show ?case by simp
- next
- case (goal3 x t y s)
- then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
- option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
- (match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
- then have
- c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
- d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
- e: "alpha_lam_raw (p \<bullet> t) s" and
- f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
- g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
- let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
- have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
- show ?case
- unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
- proof
- show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
- next
- have "atom y \<sharp> p" sorry
- have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
- then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" using swap_fresh by auto
- then have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
- have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
- then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
- then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
- ((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
- qed
- qed
- qed
-
-lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
-*)
-(*
-lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
-by (induct x rule: lam.induct) (simp_all add: match_App_simps)
-
-lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
- apply (induct x rule: lam.induct)
- apply (simp_all add: match_Lam_simps)
- apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
- apply (simp add: match_Lam_def)
- apply (subgoal_tac "\<exists>n s. Lam name lam = Lam n s")
- prefer 2
- apply auto[1]
- apply (simp add: Let_def)
- apply (thin_tac "\<exists>n s. Lam name lam = Lam n s")
- apply clarify
- apply (rule conjI)
- apply (rule_tac t="THE s. Lam name lam = Lam (new (S, Lam name lam)) s" and
- s="(name \<leftrightarrow> (new (S, Lam name lam))) \<bullet> lam" in subst)
- defer
- apply (simp add: lam.eq_iff)
- apply (rule_tac x="(name \<leftrightarrow> (new (S, Lam name lam)))" in exI)
- apply (simp add: alphas)
- apply (simp add: eqvts)
- apply (rule conjI)
- sorry
-
-function subst where
-"subst v s t = (
- case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
- case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
- case match_Lam (v,s) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
-by pat_completeness auto
-
-termination apply (relation "measure (\<lambda>(_, _, t). size t)")
- apply auto[1]
- apply (case_tac a) apply simp
- apply (frule lam_some) apply simp
- apply (case_tac a) apply simp
- apply (frule app_some) apply simp
- apply (case_tac a) apply simp
- apply (frule app_some) apply simp
-done
-
-lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
-
-lemma subst_eqvt:
- "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
- proof (induct v s t rule: subst.induct)
- case (1 v s t)
- show ?case proof (cases t rule: lam_exhaust)
- fix n
- assume "t = Var n"
- then show ?thesis by (simp add: match_Var_simps)
- next
- fix l r
- assume "t = App l r"
- then show ?thesis
- apply (simp only:)
- apply (subst subst.simps)
- apply (subst match_Var_simps)
- apply (simp only: option.cases)
- apply (subst match_App_simps)
- apply (simp only: option.cases)
- apply (simp only: prod.cases)
- apply (simp only: lam.perm)
- apply (subst (3) subst.simps)
- apply (subst match_Var_simps)
- apply (simp only: option.cases)
- apply (subst match_App_simps)
- apply (simp only: option.cases)
- apply (simp only: prod.cases)
- apply (subst 1(2)[of "(l, r)" "l" "r"])
- apply (simp add: match_Var_simps)
- apply (simp add: match_App_simps)
- apply (rule refl)
- apply (subst 1(3)[of "(l, r)" "l" "r"])
- apply (simp add: match_Var_simps)
- apply (simp add: match_App_simps)
- apply (rule refl)
- apply (rule refl)
- done
- next
- fix n t'
- assume "t = Lam n t'"
- then show ?thesis
- apply (simp only: )
- apply (simp only: lam.perm)
- apply (subst subst.simps)
- apply (subst match_Var_simps)
- apply (simp only: option.cases)
- apply (subst match_App_simps)
- apply (simp only: option.cases)
- apply (rule_tac t="Lam n t'" and s="Lam (new ((v, s), Lam n t')) ((n \<leftrightarrow> new ((v, s), Lam n t')) \<bullet> t')" in subst)
- defer
- apply (subst match_Lam_simps)
- defer
- apply (simp only: option.cases)
- apply (simp only: prod.cases)
- apply (subst (2) subst.simps)
- apply (subst match_Var_simps)
- apply (simp only: option.cases)
- apply (subst match_App_simps)
- apply (simp only: option.cases)
- apply (rule_tac t="Lam (p \<bullet> n) (p \<bullet> t')" and s="Lam (new ((p \<bullet> v, p \<bullet> s), Lam (p \<bullet> n) (p \<bullet> t'))) (((p \<bullet> n) \<leftrightarrow> new ((p \<bullet> v, p \<bullet> s), Lam (p \<bullet> n) (p \<bullet> t'))) \<bullet> t')" in subst)
- defer
- apply (subst match_Lam_simps)
- defer
- apply (simp only: option.cases)
- apply (simp only: prod.cases)
- apply (simp only: lam.perm)
- thm 1(1)
- sorry
- qed
- qed
-
-lemma subst_proper_eqs:
- "subst y s (Var x) = (if x = y then s else (Var x))"
- "subst y s (App l r) = App (subst y s l) (subst y s r)"
- "atom x \<sharp> (t, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"
- apply (subst subst.simps)
- apply (simp only: match_Var_simps)
- apply (simp only: option.simps)
- apply (subst subst.simps)
- apply (simp only: match_App_simps)
- apply (simp only: option.simps)
- apply (simp only: prod.simps)
- apply (simp only: match_Var_simps)
- apply (simp only: option.simps)
- apply (subst subst.simps)
- apply (simp only: match_Var_simps)
- apply (simp only: option.simps)
- apply (simp only: match_App_simps)
- apply (simp only: option.simps)
- apply (rule_tac t="Lam x t" and s="Lam (new ((y, s), Lam x t)) ((x \<leftrightarrow> new ((y, s), Lam x t)) \<bullet> t)" in subst)
- defer
- apply (subst match_Lam_simps)
- defer
- apply (simp only: option.simps)
- apply (simp only: prod.simps)
- sorry
-*)
end