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theory WeakNorm(* Title: HOL/Lambda/WeakNorm.thy ID: $Id: WeakNorm.thy,v 1.52 2007/10/12 06:21:10 haftmann Exp $ Author: Stefan Berghofer Copyright 2003 TU Muenchen *) header {* Weak normalization for simply-typed lambda calculus *} theory WeakNorm imports Type NormalForm Code_Integer begin text {* Formalization by Stefan Berghofer. Partly based on a paper proof by Felix Joachimski and Ralph Matthes \cite{Matthes-Joachimski-AML}. *} subsection {* Main theorems *} lemma norm_list: assumes f_compat: "!!t t'. t ->β* t' ==> f t ->β* f t'" and f_NF: "!!t. NF t ==> NF (f t)" and uNF: "NF u" and uT: "e \<turnstile> u : T" shows "!!Us. e〈i:T〉 \<tturnstile> as : Us ==> listall (λt. ∀e T' u i. e〈i:T〉 \<turnstile> t : T' --> NF u --> e \<turnstile> u : T --> (∃t'. t[u/i] ->β* t' ∧ NF t')) as ==> ∃as'. ∀j. Var j °° map (λt. f (t[u/i])) as ->β* Var j °° map f as' ∧ NF (Var j °° map f as')" (is "!!Us. _ ==> listall ?R as ==> ∃as'. ?ex Us as as'") proof (induct as rule: rev_induct) case (Nil Us) with Var_NF have "?ex Us [] []" by simp thus ?case .. next case (snoc b bs Us) have "e〈i:T〉 \<tturnstile> bs @ [b] : Us" by fact then obtain Vs W where Us: "Us = Vs @ [W]" and bs: "e〈i:T〉 \<tturnstile> bs : Vs" and bT: "e〈i:T〉 \<turnstile> b : W" by (rule types_snocE) from snoc have "listall ?R bs" by simp with bs have "∃bs'. ?ex Vs bs bs'" by (rule snoc) then obtain bs' where bsred: "!!j. Var j °° map (λt. f (t[u/i])) bs ->β* Var j °° map f bs'" and bsNF: "!!j. NF (Var j °° map f bs')" by iprover from snoc have "?R b" by simp with bT and uNF and uT have "∃b'. b[u/i] ->β* b' ∧ NF b'" by iprover then obtain b' where bred: "b[u/i] ->β* b'" and bNF: "NF b'" by iprover from bsNF [of 0] have "listall NF (map f bs')" by (rule App_NF_D) moreover have "NF (f b')" using bNF by (rule f_NF) ultimately have "listall NF (map f (bs' @ [b']))" by simp hence "!!j. NF (Var j °° map f (bs' @ [b']))" by (rule NF.App) moreover from bred have "f (b[u/i]) ->β* f b'" by (rule f_compat) with bsred have "!!j. (Var j °° map (λt. f (t[u/i])) bs) ° f (b[u/i]) ->β* (Var j °° map f bs') ° f b'" by (rule rtrancl_beta_App) ultimately have "?ex Us (bs @ [b]) (bs' @ [b'])" by simp thus ?case .. qed lemma subst_type_NF: "!!t e T u i. NF t ==> e〈i:U〉 \<turnstile> t : T ==> NF u ==> e \<turnstile> u : U ==> ∃t'. t[u/i] ->β* t' ∧ NF t'" (is "PROP ?P U" is "!!t e T u i. _ ==> PROP ?Q t e T u i U") proof (induct U) fix T t let ?R = "λt. ∀e T' u i. e〈i:T〉 \<turnstile> t : T' --> NF u --> e \<turnstile> u : T --> (∃t'. t[u/i] ->β* t' ∧ NF t')" assume MI1: "!!T1 T2. T = T1 => T2 ==> PROP ?P T1" assume MI2: "!!T1 T2. T = T1 => T2 ==> PROP ?P T2" assume "NF t" thus "!!e T' u i. PROP ?Q t e T' u i T" proof induct fix e T' u i assume uNF: "NF u" and uT: "e \<turnstile> u : T" { case (App ts x e_ T'_ u_ i_) assume "e〈i:T〉 \<turnstile> Var x °° ts : T'" then obtain Us where varT: "e〈i:T〉 \<turnstile> Var x : Us \<Rrightarrow> T'" and argsT: "e〈i:T〉 \<tturnstile> ts : Us" by (rule var_app_typesE) from nat_eq_dec show "∃t'. (Var x °° ts)[u/i] ->β* t' ∧ NF t'" proof assume eq: "x = i" show ?thesis proof (cases ts) case Nil with eq have "(Var x °° [])[u/i] ->β* u" by simp with Nil and uNF show ?thesis by simp iprover next case (Cons a as) with argsT obtain T'' Ts where Us: "Us = T'' # Ts" by (cases Us) (rule FalseE, simp+, erule that) from varT and Us have varT: "e〈i:T〉 \<turnstile> Var x : T'' => Ts \<Rrightarrow> T'" by simp from varT eq have T: "T = T'' => Ts \<Rrightarrow> T'" by cases auto with uT have uT': "e \<turnstile> u : T'' => Ts \<Rrightarrow> T'" by simp from argsT Us Cons have argsT': "e〈i:T〉 \<tturnstile> as : Ts" by simp from argsT Us Cons have argT: "e〈i:T〉 \<turnstile> a : T''" by simp from argT uT refl have aT: "e \<turnstile> a[u/i] : T''" by (rule subst_lemma) from App and Cons have "listall ?R as" by simp (iprover dest: listall_conj2) with lift_preserves_beta' lift_NF uNF uT argsT' have "∃as'. ∀j. Var j °° map (λt. lift (t[u/i]) 0) as ->β* Var j °° map (λt. lift t 0) as' ∧ NF (Var j °° map (λt. lift t 0) as')" by (rule norm_list) then obtain as' where asred: "Var 0 °° map (λt. lift (t[u/i]) 0) as ->β* Var 0 °° map (λt. lift t 0) as'" and asNF: "NF (Var 0 °° map (λt. lift t 0) as')" by iprover from App and Cons have "?R a" by simp with argT and uNF and uT have "∃a'. a[u/i] ->β* a' ∧ NF a'" by iprover then obtain a' where ared: "a[u/i] ->β* a'" and aNF: "NF a'" by iprover from uNF have "NF (lift u 0)" by (rule lift_NF) hence "∃u'. lift u 0 ° Var 0 ->β* u' ∧ NF u'" by (rule app_Var_NF) then obtain u' where ured: "lift u 0 ° Var 0 ->β* u'" and u'NF: "NF u'" by iprover from T and u'NF have "∃ua. u'[a'/0] ->β* ua ∧ NF ua" proof (rule MI1) have "e〈0:T''〉 \<turnstile> lift u 0 ° Var 0 : Ts \<Rrightarrow> T'" proof (rule typing.App) from uT' show "e〈0:T''〉 \<turnstile> lift u 0 : T'' => Ts \<Rrightarrow> T'" by (rule lift_type) show "e〈0:T''〉 \<turnstile> Var 0 : T''" by (rule typing.Var) simp qed with ured show "e〈0:T''〉 \<turnstile> u' : Ts \<Rrightarrow> T'" by (rule subject_reduction') from ared aT show "e \<turnstile> a' : T''" by (rule subject_reduction') show "NF a'" by fact qed then obtain ua where uared: "u'[a'/0] ->β* ua" and uaNF: "NF ua" by iprover from ared have "(lift u 0 ° Var 0)[a[u/i]/0] ->β* (lift u 0 ° Var 0)[a'/0]" by (rule subst_preserves_beta2') also from ured have "(lift u 0 ° Var 0)[a'/0] ->β* u'[a'/0]" by (rule subst_preserves_beta') also note uared finally have "(lift u 0 ° Var 0)[a[u/i]/0] ->β* ua" . hence uared': "u ° a[u/i] ->β* ua" by simp from T asNF _ uaNF have "∃r. (Var 0 °° map (λt. lift t 0) as')[ua/0] ->β* r ∧ NF r" proof (rule MI2) have "e〈0:Ts \<Rrightarrow> T'〉 \<turnstile> Var 0 °° map (λt. lift (t[u/i]) 0) as : T'" proof (rule list_app_typeI) show "e〈0:Ts \<Rrightarrow> T'〉 \<turnstile> Var 0 : Ts \<Rrightarrow> T'" by (rule typing.Var) simp from uT argsT' have "e \<tturnstile> map (λt. t[u/i]) as : Ts" by (rule substs_lemma) hence "e〈0:Ts \<Rrightarrow> T'〉 \<tturnstile> map (λt. lift t 0) (map (λt. t[u/i]) as) : Ts" by (rule lift_types) thus "e〈0:Ts \<Rrightarrow> T'〉 \<tturnstile> map (λt. lift (t[u/i]) 0) as : Ts" by (simp_all add: map_compose [symmetric] o_def) qed with asred show "e〈0:Ts \<Rrightarrow> T'〉 \<turnstile> Var 0 °° map (λt. lift t 0) as' : T'" by (rule subject_reduction') from argT uT refl have "e \<turnstile> a[u/i] : T''" by (rule subst_lemma) with uT' have "e \<turnstile> u ° a[u/i] : Ts \<Rrightarrow> T'" by (rule typing.App) with uared' show "e \<turnstile> ua : Ts \<Rrightarrow> T'" by (rule subject_reduction') qed then obtain r where rred: "(Var 0 °° map (λt. lift t 0) as')[ua/0] ->β* r" and rnf: "NF r" by iprover from asred have "(Var 0 °° map (λt. lift (t[u/i]) 0) as)[u ° a[u/i]/0] ->β* (Var 0 °° map (λt. lift t 0) as')[u ° a[u/i]/0]" by (rule subst_preserves_beta') also from uared' have "(Var 0 °° map (λt. lift t 0) as')[u ° a[u/i]/0] ->β* (Var 0 °° map (λt. lift t 0) as')[ua/0]" by (rule subst_preserves_beta2') also note rred finally have "(Var 0 °° map (λt. lift (t[u/i]) 0) as)[u ° a[u/i]/0] ->β* r" . with rnf Cons eq show ?thesis by (simp add: map_compose [symmetric] o_def) iprover qed next assume neq: "x ≠ i" from App have "listall ?R ts" by (iprover dest: listall_conj2) with TrueI TrueI uNF uT argsT have "∃ts'. ∀j. Var j °° map (λt. t[u/i]) ts ->β* Var j °° ts' ∧ NF (Var j °° ts')" (is "∃ts'. ?ex ts'") by (rule norm_list [of "λt. t", simplified]) then obtain ts' where NF: "?ex ts'" .. from nat_le_dec show ?thesis proof assume "i < x" with NF show ?thesis by simp iprover next assume "¬ (i < x)" with NF neq show ?thesis by (simp add: subst_Var) iprover qed qed next case (Abs r e_ T'_ u_ i_) assume absT: "e〈i:T〉 \<turnstile> Abs r : T'" then obtain R S where "e〈0:R〉〈Suc i:T〉 \<turnstile> r : S" by (rule abs_typeE) simp moreover have "NF (lift u 0)" using `NF u` by (rule lift_NF) moreover have "e〈0:R〉 \<turnstile> lift u 0 : T" using uT by (rule lift_type) ultimately have "∃t'. r[lift u 0/Suc i] ->β* t' ∧ NF t'" by (rule Abs) thus "∃t'. Abs r[u/i] ->β* t' ∧ NF t'" by simp (iprover intro: rtrancl_beta_Abs NF.Abs) } qed qed -- {* A computationally relevant copy of @{term "e \<turnstile> t : T"} *} inductive rtyping :: "(nat => type) => dB => type => bool" ("_ \<turnstile>R _ : _" [50, 50, 50] 50) where Var: "e x = T ==> e \<turnstile>R Var x : T" | Abs: "e〈0:T〉 \<turnstile>R t : U ==> e \<turnstile>R Abs t : (T => U)" | App: "e \<turnstile>R s : T => U ==> e \<turnstile>R t : T ==> e \<turnstile>R (s ° t) : U" lemma rtyping_imp_typing: "e \<turnstile>R t : T ==> e \<turnstile> t : T" apply (induct set: rtyping) apply (erule typing.Var) apply (erule typing.Abs) apply (erule typing.App) apply assumption done theorem type_NF: assumes "e \<turnstile>R t : T" shows "∃t'. t ->β* t' ∧ NF t'" using assms proof induct case Var show ?case by (iprover intro: Var_NF) next case Abs thus ?case by (iprover intro: rtrancl_beta_Abs NF.Abs) next case (App e s T U t) from App obtain s' t' where sred: "s ->β* s'" and "NF s'" and tred: "t ->β* t'" and tNF: "NF t'" by iprover have "∃u. (Var 0 ° lift t' 0)[s'/0] ->β* u ∧ NF u" proof (rule subst_type_NF) have "NF (lift t' 0)" using tNF by (rule lift_NF) hence "listall NF [lift t' 0]" by (rule listall_cons) (rule listall_nil) hence "NF (Var 0 °° [lift t' 0])" by (rule NF.App) thus "NF (Var 0 ° lift t' 0)" by simp show "e〈0:T => U〉 \<turnstile> Var 0 ° lift t' 0 : U" proof (rule typing.App) show "e〈0:T => U〉 \<turnstile> Var 0 : T => U" by (rule typing.Var) simp from tred have "e \<turnstile> t' : T" by (rule subject_reduction') (rule rtyping_imp_typing, rule App.hyps) thus "e〈0:T => U〉 \<turnstile> lift t' 0 : T" by (rule lift_type) qed from sred show "e \<turnstile> s' : T => U" by (rule subject_reduction') (rule rtyping_imp_typing, rule App.hyps) show "NF s'" by fact qed then obtain u where ured: "s' ° t' ->β* u" and unf: "NF u" by simp iprover from sred tred have "s ° t ->β* s' ° t'" by (rule rtrancl_beta_App) hence "s ° t ->β* u" using ured by (rule rtranclp_trans) with unf show ?case by iprover qed subsection {* Extracting the program *} declare NF.induct [ind_realizer] declare rtranclp.induct [ind_realizer irrelevant] declare rtyping.induct [ind_realizer] lemmas [extraction_expand] = conj_assoc listall_cons_eq extract type_NF lemma rtranclR_rtrancl_eq: "rtranclpR r a b = r** a b" apply (rule iffI) apply (erule rtranclpR.induct) apply (rule rtranclp.rtrancl_refl) apply (erule rtranclp.rtrancl_into_rtrancl) apply assumption apply (erule rtranclp.induct) apply (rule rtranclpR.rtrancl_refl) apply (erule rtranclpR.rtrancl_into_rtrancl) apply assumption done lemma NFR_imp_NF: "NFR nf t ==> NF t" apply (erule NFR.induct) apply (rule NF.intros) apply (simp add: listall_def) apply (erule NF.intros) done text_raw {* \begin{figure} \renewcommand{\isastyle}{\scriptsize\it}% @{thm [display,eta_contract=false,margin=100] subst_type_NF_def} \renewcommand{\isastyle}{\small\it}% \caption{Program extracted from @{text subst_type_NF}} \label{fig:extr-subst-type-nf} \end{figure} \begin{figure} \renewcommand{\isastyle}{\scriptsize\it}% @{thm [display,margin=100] subst_Var_NF_def} @{thm [display,margin=100] app_Var_NF_def} @{thm [display,margin=100] lift_NF_def} @{thm [display,eta_contract=false,margin=100] type_NF_def} \renewcommand{\isastyle}{\small\it}% \caption{Program extracted from lemmas and main theorem} \label{fig:extr-type-nf} \end{figure} *} text {* The program corresponding to the proof of the central lemma, which performs substitution and normalization, is shown in Figure \ref{fig:extr-subst-type-nf}. The correctness theorem corresponding to the program @{text "subst_type_NF"} is @{thm [display,margin=100] subst_type_NF_correctness [simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]} where @{text NFR} is the realizability predicate corresponding to the datatype @{text NFT}, which is inductively defined by the rules \pagebreak @{thm [display,margin=90] NFR.App [of ts nfs x] NFR.Abs [of nf t]} The programs corresponding to the main theorem @{text "type_NF"}, as well as to some lemmas, are shown in Figure \ref{fig:extr-type-nf}. The correctness statement for the main function @{text "type_NF"} is @{thm [display,margin=100] type_NF_correctness [simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]} where the realizability predicate @{text "rtypingR"} corresponding to the computationally relevant version of the typing judgement is inductively defined by the rules @{thm [display,margin=100] rtypingR.Var [no_vars] rtypingR.Abs [of ty, no_vars] rtypingR.App [of ty e s T U ty' t]} *} subsection {* Generating executable code *} consts_code "arbitrary :: 'a" ("(error \"arbitrary\")") "arbitrary :: 'a => 'b" ("(fn '_ => error \"arbitrary\")") code_module Norm contains test = "type_NF" text {* The following functions convert between Isabelle's built-in {\tt term} datatype and the generated {\tt dB} datatype. This allows to generate example terms using Isabelle's parser and inspect normalized terms using Isabelle's pretty printer. *} ML {* fun nat_of_int 0 = Norm.zero | nat_of_int n = Norm.Suc (nat_of_int (n-1)); fun int_of_nat Norm.zero = 0 | int_of_nat (Norm.Suc n) = 1 + int_of_nat n; fun dBtype_of_typ (Type ("fun", [T, U])) = Norm.Fun (dBtype_of_typ T, dBtype_of_typ U) | dBtype_of_typ (TFree (s, _)) = (case explode s of ["'", a] => Norm.Atom (nat_of_int (ord a - 97)) | _ => error "dBtype_of_typ: variable name") | dBtype_of_typ _ = error "dBtype_of_typ: bad type"; fun dB_of_term (Bound i) = Norm.dB_Var (nat_of_int i) | dB_of_term (t $ u) = Norm.App (dB_of_term t, dB_of_term u) | dB_of_term (Abs (_, _, t)) = Norm.Abs (dB_of_term t) | dB_of_term _ = error "dB_of_term: bad term"; fun term_of_dB Ts (Type ("fun", [T, U])) (Norm.Abs dBt) = Abs ("x", T, term_of_dB (T :: Ts) U dBt) | term_of_dB Ts _ dBt = term_of_dB' Ts dBt and term_of_dB' Ts (Norm.dB_Var n) = Bound (int_of_nat n) | term_of_dB' Ts (Norm.App (dBt, dBu)) = let val t = term_of_dB' Ts dBt in case fastype_of1 (Ts, t) of Type ("fun", [T, U]) => t $ term_of_dB Ts T dBu | _ => error "term_of_dB: function type expected" end | term_of_dB' _ _ = error "term_of_dB: term not in normal form"; fun typing_of_term Ts e (Bound i) = Norm.Var (e, nat_of_int i, dBtype_of_typ (List.nth (Ts, i))) | typing_of_term Ts e (t $ u) = (case fastype_of1 (Ts, t) of Type ("fun", [T, U]) => Norm.rtypingT_App (e, dB_of_term t, dBtype_of_typ T, dBtype_of_typ U, dB_of_term u, typing_of_term Ts e t, typing_of_term Ts e u) | _ => error "typing_of_term: function type expected") | typing_of_term Ts e (Abs (s, T, t)) = let val dBT = dBtype_of_typ T in Norm.rtypingT_Abs (e, dBT, dB_of_term t, dBtype_of_typ (fastype_of1 (T :: Ts, t)), typing_of_term (T :: Ts) (Norm.shift e Norm.zero dBT) t) end | typing_of_term _ _ _ = error "typing_of_term: bad term"; fun dummyf _ = error "dummy"; *} text {* We now try out the extracted program @{text "type_NF"} on some example terms. *} ML {* val ct1 = @{cterm "%f. ((%f x. f (f (f x))) ((%f x. f (f (f (f x)))) f))"}; val (dB1, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct1)); val ct1' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct1)) dB1); val ct2 = @{cterm "%f x. (%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) x)))))"}; val (dB2, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct2)); val ct2' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct2)) dB2); *} text {* The same story again for code next generation. *} setup {* CodeTarget.add_undefined "SML" "arbitrary" "(raise Fail \"arbitrary\")" *} definition int_of_nat :: "nat => int" where "int_of_nat = of_nat" export_code type_NF nat int_of_nat in SML module_name Norm ML {* val nat_of_int = Norm.nat; val int_of_nat = Norm.int_of_nat; fun dBtype_of_typ (Type ("fun", [T, U])) = Norm.Fun (dBtype_of_typ T, dBtype_of_typ U) | dBtype_of_typ (TFree (s, _)) = (case explode s of ["'", a] => Norm.Atom (nat_of_int (ord a - 97)) | _ => error "dBtype_of_typ: variable name") | dBtype_of_typ _ = error "dBtype_of_typ: bad type"; fun dB_of_term (Bound i) = Norm.Var (nat_of_int i) | dB_of_term (t $ u) = Norm.App (dB_of_term t, dB_of_term u) | dB_of_term (Abs (_, _, t)) = Norm.Abs (dB_of_term t) | dB_of_term _ = error "dB_of_term: bad term"; fun term_of_dB Ts (Type ("fun", [T, U])) (Norm.Abs dBt) = Abs ("x", T, term_of_dB (T :: Ts) U dBt) | term_of_dB Ts _ dBt = term_of_dB' Ts dBt and term_of_dB' Ts (Norm.Var n) = Bound (int_of_nat n) | term_of_dB' Ts (Norm.App (dBt, dBu)) = let val t = term_of_dB' Ts dBt in case fastype_of1 (Ts, t) of Type ("fun", [T, U]) => t $ term_of_dB Ts T dBu | _ => error "term_of_dB: function type expected" end | term_of_dB' _ _ = error "term_of_dB: term not in normal form"; fun typing_of_term Ts e (Bound i) = Norm.Vara (e, nat_of_int i, dBtype_of_typ (nth Ts i)) | typing_of_term Ts e (t $ u) = (case fastype_of1 (Ts, t) of Type ("fun", [T, U]) => Norm.Appb (e, dB_of_term t, dBtype_of_typ T, dBtype_of_typ U, dB_of_term u, typing_of_term Ts e t, typing_of_term Ts e u) | _ => error "typing_of_term: function type expected") | typing_of_term Ts e (Abs (s, T, t)) = let val dBT = dBtype_of_typ T in Norm.Absb (e, dBT, dB_of_term t, dBtype_of_typ (fastype_of1 (T :: Ts, t)), typing_of_term (T :: Ts) (Norm.shift e Norm.Zero_nat dBT) t) end | typing_of_term _ _ _ = error "typing_of_term: bad term"; fun dummyf _ = error "dummy"; *} ML {* val ct1 = @{cterm "%f. ((%f x. f (f (f x))) ((%f x. f (f (f (f x)))) f))"}; val (dB1, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct1)); val ct1' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct1)) dB1); val ct2 = @{cterm "%f x. (%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) x)))))"}; val (dB2, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct2)); val ct2' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct2)) dB2); *} end
lemma norm_list:
(!!t t'. t ->> t' ==> f t ->> f t')
==> (!!t. NF t ==> NF (f t))
==> NF u
==> e \<turnstile> u : T
==> e〈i:T〉 ||- as : Us
==> listall
(λt. ∀e T' u i.
e〈i:T〉 \<turnstile> t : T' -->
NF u -->
e \<turnstile> u : T -->
(∃t'. t[u/i] ->> t' ∧ NF t'))
as
==> ∃as'. ∀j. Var j °° map (λt. f (t[u/i])) as ->>
Var j °° map f as' ∧
NF (Var j °° map f as')
lemma subst_type_NF:
NF t
==> e〈i:U〉 \<turnstile> t : T
==> NF u ==> e \<turnstile> u : U ==> ∃t'. t[u/i] ->> t' ∧ NF t'
lemma rtyping_imp_typing:
e \<turnstile>R t : T ==> e \<turnstile> t : T
theorem type_NF:
e \<turnstile>R t : T ==> ∃t'. t ->> t' ∧ NF t'
lemma
((P ∧ Q) ∧ R) = (P ∧ Q ∧ R)
listall P (x # xs) = (P x ∧ listall P xs)
lemma rtranclR_rtrancl_eq:
rtranclpR r a b = r** a b
lemma NFR_imp_NF:
NFR nf t ==> NF t