475 (* Substitution *)

   476 

   477 definition new where

   478   "new s = (THE x. \<forall>a \<in> s. atom x \<noteq> a)"

   479 

   480 lemma size_no_change: "size (p \<bullet> (t :: lam_raw)) = size t"

   481   by (induct t) simp_all

   482 

   483 function

   484   subst_raw :: "lam_raw \<Rightarrow> name \<Rightarrow> lam_raw \<Rightarrow> lam_raw"

   485 where

   486   "subst_raw (Var_raw x) y s = (if x=y then s else (Var_raw x))"

   487 | "subst_raw (App_raw l r) y s = App_raw (subst_raw l y s) (subst_raw r y s)"

   488 | "subst_raw (Lam_raw x t) y s =

   489       Lam_raw (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))

   490        (subst_raw ((x \<leftrightarrow> (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))) \<bullet> t) y s)"

   491 by (pat_completeness, auto)

   492 termination

   493   apply (relation "measure (\<lambda>(t, y, s). (size t))")

   494   apply (auto simp add: size_no_change)

   495   done

   496 

   497 lemma fv_subst[simp]: "fv_lam_raw (subst_raw t y s) =

   498   (if (atom y \<in> fv_lam_raw t) then fv_lam_raw s \<union> (fv_lam_raw t - {atom y}) else fv_lam_raw t)"

   499   apply (induct t arbitrary: s)

   500   apply (auto simp add: supp_at_base)[1]

   501   apply (auto simp add: supp_at_base)[1]

   502   apply (simp only: fv_lam_raw.simps)

   503   apply simp

   504   apply (rule conjI)

   505   apply clarify

   506   sorry

   507 

   508 thm supp_at_base

   509 lemma new_eqvt[eqvt]: "p \<bullet> (new s) = new (p \<bullet> s)"

   510   sorry

   511 

   512 lemma subst_var_raw_eqvt[eqvt]: "p \<bullet> (subst_raw t y s) = subst_raw (p \<bullet> t) (p \<bullet> y) (p \<bullet> s)"

   513   apply (induct t arbitrary: p y s)

   514   apply simp_all

   515   apply(perm_simp)

   516   apply simp

   517   sorry

   518 

   519 lemma subst_id: "alpha_lam_raw (subst_raw x d (Var_raw d)) x"

   520   apply (induct x arbitrary: d)

   521   apply (simp_all add: alpha_lam_raw.intros)

   522   apply (rule alpha_lam_raw.intros)

   523   apply (rule_tac x="(name \<leftrightarrow> new (insert (atom d) (supp d)))" in exI)

   524   apply (simp add: alphas)

   525   oops

   526 

   527 quotient_definition

   528   subst ("_ [ _ ::= _ ]" [100,100,100] 100)

   529 where

   530   "subst :: lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" is "subst_raw"

   531 

   532 lemmas fv_rsp = quot_respect(10)[simplified]

   533 

   534 lemma subst_rsp_pre1:

   535   assumes a: "alpha_lam_raw a b"

   536   shows "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)"

   537   using a

   538   apply (induct a b arbitrary: c y rule: alpha_lam_raw.induct)

   539   apply (simp add: equivp_reflp[OF lam_equivp])

   540   apply (simp add: alpha_lam_raw.intros)

   541   apply (simp only: alphas)

   542   apply clarify

   543   apply (simp only: subst_raw.simps)

   544   apply (rule alpha_lam_raw.intros)

   545   apply (simp only: alphas)

   546   sorry

   547 

   548 lemma subst_rsp_pre2:

   549   assumes a: "alpha_lam_raw a b"

   550   shows "alpha_lam_raw (subst_raw c y a) (subst_raw c y b)"

   551   using a

   552   apply (induct c arbitrary: a b y)

   553   apply (simp add: equivp_reflp[OF lam_equivp])

   554   apply (simp add: alpha_lam_raw.intros)

   555   apply simp

   556   apply (rule alpha_lam_raw.intros)

   557   sorry

   558 

   559 lemma [quot_respect]:

   560   "(alpha_lam_raw ===> op = ===> alpha_lam_raw ===> alpha_lam_raw) subst_raw subst_raw"

   561   proof (intro fun_relI, simp)

   562     fix a b c d :: lam_raw

   563     fix y :: name

   564     assume a: "alpha_lam_raw a b"

   565     assume b: "alpha_lam_raw c d"

   566     have c: "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)" using subst_rsp_pre1 a by simp

   567     then have d: "alpha_lam_raw (subst_raw b y c) (subst_raw b y d)" using subst_rsp_pre2 b by simp

   568     show "alpha_lam_raw (subst_raw a y c) (subst_raw b y d)"

   569       using c d equivp_transp[OF lam_equivp] by blast

   570   qed

   571 

   572 lemma simp3:

   573   "x \<noteq> y \<Longrightarrow> atom x \<notin> fv_lam_raw s \<Longrightarrow> alpha_lam_raw (subst_raw (Lam_raw x t) y s) (Lam_raw x (subst_raw t y s))"

   574   apply simp

   575   apply (rule alpha_lam_raw.intros)

   576   apply (rule_tac x ="(x \<leftrightarrow> (new (insert (atom y) (fv_lam_raw s \<union> fv_lam_raw t) -

   577                     {atom x})))" in exI)

   578   apply (simp only: alphas)

   579   apply simp

   580   sorry

   581 

   582 lemmas subst_simps = subst_raw.simps(1-2)[quot_lifted,no_vars]

   583   simp3[quot_lifted,simplified lam.supp,simplified fresh_def[symmetric], no_vars]