From 3131bba5ab61966ab34a439f8d02268a0ab2c284 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Wed, 24 Dec 2014 09:56:26 +0000 Subject: [PATCH] finite lambda calculus --- matita/matita/lib/finite_lambda/confluence.ma | 226 +++++++ .../matita/lib/finite_lambda/finite_lambda.ma | 456 ++++++++++++++ .../lib/finite_lambda/finite_lambda_deep.ma | 562 ++++++++++++++++++ matita/matita/lib/finite_lambda/reduction.ma | 308 ++++++++++ .../lib/finite_lambda/terms_and_types.ma | 326 ++++++++++ matita/matita/lib/finite_lambda/typing.ma | 229 +++++++ .../matita/lib/finite_lambda/typing_deep.ma | 239 ++++++++ matita/matita/lib/finite_lambda/typing_old.ma | 236 ++++++++ 8 files changed, 2582 insertions(+) create mode 100644 matita/matita/lib/finite_lambda/confluence.ma create mode 100644 matita/matita/lib/finite_lambda/finite_lambda.ma create mode 100644 matita/matita/lib/finite_lambda/finite_lambda_deep.ma create mode 100644 matita/matita/lib/finite_lambda/reduction.ma create mode 100644 matita/matita/lib/finite_lambda/terms_and_types.ma create mode 100644 matita/matita/lib/finite_lambda/typing.ma create mode 100644 matita/matita/lib/finite_lambda/typing_deep.ma create mode 100644 matita/matita/lib/finite_lambda/typing_old.ma diff --git a/matita/matita/lib/finite_lambda/confluence.ma b/matita/matita/lib/finite_lambda/confluence.ma new file mode 100644 index 000000000..32b7e3ff0 --- /dev/null +++ b/matita/matita/lib/finite_lambda/confluence.ma @@ -0,0 +1,226 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "finite_lambda/reduction.ma". + + +axiom canonical_to_T: ∀O,D.∀M:T O D.∀ty.(* type_of M ty → *) + ∃a:FinSet_of_FType O D ty. star ? (red O D) M (to_T O D ty a). + +axiom normal_to_T: ∀O,D,M,ty,a. red O D (to_T O D ty a) M → False. + +axiom red_closed: ∀O,D,M,M1. + is_closed O D 0 M → red O D M M1 → is_closed O D 0 M1. + +lemma critical: ∀O,D,ty,M,N. + ∃M3:T O D + .star (T O D) (red O D) (subst O D M 0 N) M3 + ∧star (T O D) (red O D) + (App O D + (Vec O D ty + (map (FinSet_of_FType O D ty) (T O D) + (λa0:FinSet_of_FType O D ty.subst O D M 0 (to_T O D ty a0)) + (enum (FinSet_of_FType O D ty)))) N) M3. +#O #D #ty #M #N +lapply (canonical_to_T O D N ty) * #a #Ha +%{(subst O D M 0 (to_T O D ty a))} (* CR-term *) +%[@red_star_subst @Ha + |@trans_star [|@(star_red_appr … Ha)] @R_to_star @riota + lapply (enum_complete (FinSet_of_FType O D ty) a) + elim (enum (FinSet_of_FType O D ty)) + [normalize #H1 destruct (H1) + |#hd #tl #Hind #H cases (orb_true_l … H) -H #Hcase + [normalize >Hcase >(\P Hcase) // + |normalize cases (true_or_false (a==hd)) #Hcase1 + [normalize >Hcase1 >(\P Hcase1) // |>Hcase1 @Hind @Hcase] + ] + ] + ] +qed. + +lemma critical2: ∀O,D,ty,a,M,M1,M2,v. + red O D (Vec O D ty v) M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M1 → + assoc (FinSet_of_FType O D ty) (T O D) a (enum (FinSet_of_FType O D ty)) v + =Some (T O D) M2 → + ∃M3:T O D + .star (T O D) (red O D) M2 M3 + ∧star (T O D) (red O D) (App O D M (to_T O D ty a)) M3. +#O #D #ty #a #M #M1 #M2 #v #redM #redM1 #Ha lapply (red_vec … redM) -redM +* #N * #N1 * #v1 * #v2 * * #Hred1 #Hv #HM0 >HM0 -HM0 >Hv in Ha; #Ha +cases (same_assoc … a (enum (FinSet_of_FType O D ty)) v1 v2 N N1) + [* >Ha -Ha #H1 destruct (H1) #Ha + %{N1} (* CR-term *) % [@R_to_star //|@R_to_star @(riota … Ha)] + |#Ha1 %{M2} (* CR-term *) % [// | @R_to_star @riota length_map >length_map //] #n #M0 + cases (true_or_false (leb (|enum (FinSet_of_FType O D ty)|) n)) #Hcase + [>nth_to_default [2:>length_map @(leb_true_to_le … Hcase)] + >nth_to_default [2:>length_map @(leb_true_to_le … Hcase)] // + |cut (n < |enum (FinSet_of_FType O D ty)|) + [@not_le_to_lt @leb_false_to_not_le @Hcase] #Hlt + cut (∃a:FinSet_of_FType O D ty.True) + [lapply Hlt lapply (enum_complete (FinSet_of_FType O D ty)) + cases (enum (FinSet_of_FType O D ty)) + [#_ normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#a #l #_ #_ %{a} // + ] + ] * #a #_ + >(nth_map ?????? a Hlt) >(nth_map ?????? a Hlt) #_ + @red_star_subst2 // + ] + ] +qed. + +(* we need to proceed by structural induction on the term and then +by inversion on the two redexes. The problem are the moves in a +same subterm, since we need an induction hypothesis, there *) + +lemma local_confluence: ∀O,D,M,M1,M2. red O D M M1 → red O D M M2 → +∃M3. star ? (red O D) M1 M3 ∧ star ? (red O D) M2 M3. +#O #D #M @(T_elim … M) + [#o #a #M1 #M2 #H elim(red_val ????? H) + |#n #M1 #M2 #H elim(red_rel ???? H) + |(* app : this is the interesting case *) + #P #Q #HindP #HindQ + #M1 #M2 #H1 inversion H1 -H1 + [(* right redex is beta *) + #ty #Q #N #Hc #HM >HM -HM #HM1 >HM1 - HM1 #Hl inversion Hl + [#ty1 #Q1 #N1 #Hc1 #H1 destruct (H1) #H_ + %{(subst O D Q1 0 N1)} (* CR-term *) /2/ + |#ty #v #a #M0 #_ #H1 destruct (H1) (* vacuous *) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #_ cases (red_lambda … redM0) + [* #Q1 * #redQ #HM10 >HM10 + %{(subst O D Q1 0 N0)} (* CR-term *) % + [@red_star_subst2 //|@R_to_star @rbeta @Hc] + |#HM1 >HM1 @critical + ] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #HM2 + %{(subst O D Q 0 N1)} (* CR-term *) + %[@red_star_subst @R_to_star //|@R_to_star @rbeta @(red_closed … Hc) //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is iota *)#ty #v #a #M3 #Ha #_ #_ #Hl inversion Hl + [#P1 #M1 #N1 #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #v1 #a1 #M4 #Ha1 #H1 destruct (H1) -H1 #HM4 >(inj_to_T … e0) in Ha; + >Ha1 #H1 destruct (H1) %{M3} (* CR-term *) /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 @(critical2 … redM0 Hl Ha) + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) elim (normal_to_T … redN0N1) + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appl *)#M3 #M4 #N #redM3M4 #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M1 #N1 #Hc #H1 destruct (H1) #HM2 lapply (red_lambda … redM3M4) * + [* #M3 * #H1 #H2 >H2 %{(subst O D M3 0 N1)} % + [@R_to_star @rbeta @Hc|@red_star_subst2 // ] + |#H >H -H lapply (critical O D ty1 M1 N1) * #M3 * #H1 #H2 + %{M3} /2/ + ] + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct + lapply (critical2 … redM3M4 Hl Ha1) * #M3 * #H1 #H2 %{M3} /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + lapply (HindP … redM0 redM3M4) * #M3 * #H1 #H2 + %{(App O D M3 N0)} (* CR-term *) % [@star_red_appl //|@star_red_appl //] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #_ + %{(App O D M4 N1)} % @R_to_star [@rappr //|@rappl //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appr *)#M3 #N #N1 #redN #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M0 #N0 #Hc #H1 destruct (H1) #HM2 + %{(subst O D M0 0 N1)} (* CR-term *) % + [@R_to_star @rbeta @(red_closed … Hc) //|@red_star_subst @R_to_star // ] + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct (H1) elim (normal_to_T … redN) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + %{(App O D M10 N1)} (* CR-term *) % @R_to_star [@rappl //|@rappr //] + |#M0 #N0 #N10 #redN0 #_ #H1 destruct (H1) #_ + lapply (HindQ … redN0 redN) * #M3 * #H1 #H2 + %{(App O D M0 M3)} (* CR-term *) % [@star_red_appr //|@star_red_appr //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is rlam *) #ty #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |(* right redex is rmem *) #ty #M0 #H1 destruct (H1) (* vacuous *) + |(* right redex is vec *) #ty #N #N1 #v #v1 #_ #_ + #H1 destruct (H1) (* vacuous *) + ] + |#ty #M1 #Hind #M2 #M3 #H1 #H2 (* this case is not trivial any more *) + lapply (red_lambda … H1) * + [* #M4 * #H3 #H4 >H4 lapply (red_lambda … H2) * + [* #M5 * #H5 #H6 >H6 lapply(Hind … H3 H5) * #M6 * #H7 #H8 + %{(Lambda O D ty M6)} (* CR-term *) % @star_red_lambda // + |#H5 >H5 @critical3 // + ] + |#HM2 >HM2 lapply (red_lambda … H2) * + [* #M4 * #Hred #HM3 >HM3 lapply (critical3 … ty ?? Hred) * #M5 + * #H3 #H4 %{M5} (* CR-term *) % // + |#HM3 >HM3 %{M3} (* CR-term *) % // + ] + ] + |#ty #v1 #Hind #M1 #M2 #H1 #H2 + lapply (red_vec … H1) * #N11 * #N12 * #v11 * #v12 * * #redN11 #Hv1 #HM1 + lapply (red_vec … H2) * #N21* #N22 * #v21 * #v22 * * #redN21 #Hv2 #HM2 + >Hv1 in Hv2; #Hvv lapply (compare_append … Hvv) -Hvv * + (* we must proceed by cases on the list *) * normalize + [(* N11 = N21 *) * + [>append_nil * #Hl1 #Hl2 destruct lapply(Hind N11 … redN11 redN21) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v21@M3::v12))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + |>append_nil * #Hl1 #Hl2 destruct lapply(Hind N21 … redN21 redN11) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v11@M3::v22))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + ] + |(* N11 ≠ N21 *) -Hind #P #l * + [* #Hv11 #Hv22 destruct + %{((Vec O D ty ((v21@N22::l)@N12::v12)))} (* CR-term *) % @R_to_star + [>associative_append >associative_append normalize @rvec // + |>append_cons append_cons associative_append >associative_append normalize @rvec // + ] + ] + ] + ] +qed. + + + + diff --git a/matita/matita/lib/finite_lambda/finite_lambda.ma b/matita/matita/lib/finite_lambda/finite_lambda.ma new file mode 100644 index 000000000..d07960e40 --- /dev/null +++ b/matita/matita/lib/finite_lambda/finite_lambda.ma @@ -0,0 +1,456 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "basics/finset.ma". +include "basics/star.ma". + + +inductive FType (O:Type[0]): Type[0] ≝ + | atom : O → FType O + | arrow : FType O → FType O → FType O. + +inductive T (O:Type[0]) (D:O → FinSet): Type[0] ≝ + | Val: ∀o:O.carr (D o) → T O D (* a value in a finset *) + | Rel: nat → T O D (* DB index, base is 0 *) + | App: T O D → T O D → T O D (* function, argument *) + | Lambda: FType O → T O D → T O D (* type, body *) + | Vec: FType O → list (T O D) → T O D (* type, body *) +. + +let rec FinSet_of_FType O (D:O→FinSet) (ty:FType O) on ty : FinSet ≝ + match ty with + [atom o ⇒ D o + |arrow ty1 ty2 ⇒ FinFun (FinSet_of_FType O D ty1) (FinSet_of_FType O D ty2) + ]. + +(* size *) + +let rec size O D (M:T O D) on M ≝ +match M with + [Val o a ⇒ 1 + |Rel n ⇒ 1 + |App P Q ⇒ size O D P + size O D Q + 1 + |Lambda Ty P ⇒ size O D P + 1 + |Vec Ty v ⇒ foldr ?? (λx,a. size O D x + a) 0 v +1 + ] +. + +(* axiom pos_size: ∀M. 1 ≤ size M. *) + +theorem Telim_size: ∀O,D.∀P: T O D → Prop. + (∀M. (∀N. size O D N < size O D M → P N) → P M) → ∀M. P M. +#O #D #P #H #M (cut (∀p,N. size O D N = p → P N)) + [2: /2/] +#p @(nat_elim1 p) #m #H1 #N #sizeN @H #N0 #Hlt @(H1 (size O D N0)) // +qed. + +lemma T_elim: + ∀O: Type[0].∀D:O→FinSet.∀P:T O D→Prop. + (∀o:O.∀x:D o.P (Val O D o x)) → + (∀n:ℕ.P(Rel O D n)) → + (∀m,n:T O D.P m→P n→P (App O D m n)) → + (∀Ty:FType O.∀m:T O D.P m→P(Lambda O D Ty m)) → + (∀Ty:FType O.∀v:list (T O D). + (∀x:T O D. mem ? x v → P x) → P(Vec O D Ty v)) → + ∀x:T O D.P x. +#O #D #P #Hval #Hrel #Happ #Hlam #Hvec @Telim_size #x cases x // + [ (* app *) #m #n #Hind @Happ @Hind // /2 by le_minus_to_plus/ + | (* lam *) #ty #m #Hind @Hlam @Hind normalize // + | (* vec *) #ty #v #Hind @Hvec #x lapply Hind elim v + [#Hind normalize * + |#hd #tl #Hind1 #Hind2 * + [#Hx >Hx @Hind2 normalize // + |@Hind1 #N #H @Hind2 @(lt_to_le_to_lt … H) normalize // + ] + ] + ] +qed. + + +(* arguments: k is the nesting depth (starts from 0), p is the lift *) +let rec lift O D t k p on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then Rel O D (n+p) else Rel O D n + | App m n ⇒ App O D (lift O D m k p) (lift O D n k p) + | Lambda Ty n ⇒ Lambda O D Ty (lift O D n (S k) p) + | Vec Ty v ⇒ Vec O D Ty (map ?? (λx. lift O D x k p) v) + ]. + +notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift 0 $n $M}. +notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}. + +interpretation "Lift" 'Lift n k M = (lift ?? M k n). + +let rec subst O D t k s on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then + (if (eqb k n) then lift O D s 0 n else Rel O D (n-1)) + else(Rel O D n) + | App m n ⇒ App O D (subst O D m k s) (subst O D n k s) + | Lambda T n ⇒ Lambda O D T (subst O D n (S k) s) + | Vec T v ⇒ Vec O D T (map ?? (λx. subst O D x k s) v) + ]. + +(* notation "hvbox(M break [ k ≝ N ])" + non associative with precedence 90 + for @{'Subst1 $M $k $N}. *) + +interpretation "Subst" 'Subst1 M k N = (subst M k N). + +(* closed terms ???? +let rec closed_k O D (t: T O D) k on t ≝ + match t with + [ Val o a ⇒ True + | Rel n ⇒ n < k + | App m n ⇒ (closed_k O D m k) ∧ (closed_k O D n k) + | Lambda T n ⇒ closed_k O D n (k+1) + | Vec T v ⇒ closed_list O D v k + ] + +and closed_list O D (l: list (T O D)) k on l ≝ + match l with + [ nil ⇒ True + | cons hd tl ⇒ closed_k O D hd k ∧ closed_list O D tl k + ] +. *) + +inductive is_closed (O:Type[0]) (D:O→FinSet): nat → T O D → Prop ≝ +| cval : ∀k,o,a.is_closed O D k (Val O D o a) +| cval : ∀k,n. n < k → is_closed O D k (Rel O D n) +| capp : ∀k,n,m. is_closed O D k m → is_closed O D k n → + is_closed O D k (App O D m n) +| clam : ∀T,k,m. is_closed O D (k+1) m → is_closed O D k (Lambda O D T m) +| cvec: ∀T,k,v. (∀m. mem ? m v → is_closed O D k m) → + is_closed O D k (Vec O D T v). + +lemma is_closed_rel: ∀O,D,n,k. + is_closed O D k (Rel O D n) → n < k. +#O #D #n #k #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct // + |#k0 #M #N #_ #_ #_ #H destruct + |#T #k0 #M #_ #_ #H destruct + |#T #k0 #v #_ #_ #H destruct + ] +qed. + + +(*** properties of lift and subst ***) + +lemma lift_0: ∀O,D.∀t:T O D.∀k. lift O D t k 0 = t. +#O #D #t @(T_elim … t) normalize // + [#n #k cases (leb k n) normalize // + |#o #v #Hind #k @eq_f lapply Hind -Hind elim v // + #hd #tl #Hind #Hind1 normalize @eq_f2 + [@Hind1 %1 //|@Hind #x #Hx @Hind1 %2 //] + ] +qed. + +axiom lift_closed: ∀O,D.∀t:T O D.∀k,p. + is_closed O D 0 t → lift O D t k p = t. +(* +#O #D #t @(T_elim … t) normalize // + [#n #k normalize // + |#o #v #Hind #k @eq_f lapply Hind -Hind elim v // + #hd #tl #Hind #Hind1 normalize @eq_f2 + [@Hind1 %1 //|@Hind #x #Hx @Hind1 %2 //] + ] +qed. *) + +let rec to_T O D ty on ty: FinSet_of_FType O D ty → T O D ≝ + match ty return (λty.FinSet_of_FType O D ty → T O D) with + [atom o ⇒ λa.Val O D o a + |arrow ty1 ty2 ⇒ λa:FinFun ??.Vec O D ty1 + (map ((FinSet_of_FType O D ty1)×(FinSet_of_FType O D ty2)) + (T O D) (λp.to_T O D ty2 (snd … p)) (pi1 … a)) + ] +. + +axiom inj_to_T: ∀O,D,ty,a1,a2. to_T O D ty a1 = to_T O D ty a2 → a1 = a2. + +let rec assoc (A:FinSet) (B:Type[0]) (a:A) l1 l2 on l1 : option B ≝ + match l1 with + [ nil ⇒ None ? + | cons hd1 tl1 ⇒ match l2 with + [ nil ⇒ None ? + | cons hd2 tl2 ⇒ if a==hd1 then Some ? hd2 else assoc A B a tl1 tl2 + ] + ]. + +lemma same_assoc: ∀A,B,a,l1,v1,v2,N,N1. + assoc A B a l1 (v1@N::v2) = Some ? N ∧ assoc A B a l1 (v1@N1::v2) = Some ? N1 + ∨ assoc A B a l1 (v1@N::v2) = assoc A B a l1 (v1@N1::v2). +#A #B #a #l1 #v1 #v2 #N #N1 lapply v1 -v1 elim l1 + [#v1 %2 // |#hd #tl #Hind * normalize cases (a==hd) normalize /3/] +qed. + +lemma assoc_to_mem: ∀A,B,a,l1,l2,b. + assoc A B a l1 l2 = Some ? b → mem ? b l2. +#A #B #a #l1 elim l1 + [#l2 #b normalize #H destruct + |#hd1 #tl1 #Hind * + [#b normalize #H destruct + |#hd2 #tl2 #b normalize cases (a==hd1) normalize + [#H %1 destruct //|#H %2 @Hind @H] + ] + ] +qed. + +inductive red (O:Type[0]) (D:O→FinSet) : T O D →T O D → Prop ≝ + | rbeta: ∀P,M,N. red O D (App O D (Lambda O D P M) N) (subst O D M 0 N) + | riota: ∀ty,v,a,M. + assoc (FinSet_of_FType O D ty) ? a (enum (FinSet_of_FType O D ty)) v = Some ? M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M + | rappl: ∀M,M1,N. red O D M M1 → red O D (App O D M N) (App O D M1 N) + | rappr: ∀M,N,N1. red O D N N1 → red O D (App O D M N) (App O D M N1) + | rmem: ∀ty,M. red O D (Lambda O D ty M) + (Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty)))) + | rvec: ∀ty,N,N1,v,v1. red O D N N1 → + red O D (Vec O D ty (v@N::v1)) (Vec O D ty (v@N1::v1)). + +(* some inversion cases *) +lemma red_vec: ∀O,D,ty,v,M. + red O D (Vec O D ty v) M → ∃N,N1,v1,v2. + red O D N N1 ∧ v = v1@N::v2 ∧ M = Vec O D ty (v1@N1::v2). +#O #D #ty #v #M #Hred inversion Hred + [#ty1 #M0 #N #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M0 #H destruct + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct #_ %{N} %{N1} %{v1} %{v2} /3/ + ] +qed. + +lemma red_lambda: ∀O,D,ty,M,N. + red O D (Lambda O D ty M) N → + N = Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty))). +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ // + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_val: ∀O,D,ty,a,N. + red O D (Val O D ty a) N → False. +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_rel: ∀O,D,n,N. + red O D (Rel O D n) N → False. +#O #D #n #N #Hred inversion Hred + [#ty1 #M0 #N #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma star_red_appl: ∀O,D,M,M1,N. star ? (red O D) M M1 → + star ? (red O D) (App O D M N) (App O D M1 N). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappl @HPQ +qed. + +lemma star_red_appr: ∀O,D,M,N,N1. star ? (red O D) N N1 → + star ? (red O D) (App O D M N) (App O D M N1). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappr @HPQ +qed. + +lemma star_red_vec: ∀O,D,ty,N,N1,v1,v2. star ? (red O D) N N1 → + star ? (red O D) (Vec O D ty (v1@N::v2)) (Vec O D ty (v1@N1::v2)). +#O #D #ty #N #N1 #v1 #v2 #H elim H // +#P #Q #Hind #HPQ #Hvec %1[|@Hvec] @rvec @HPQ +qed. + +axiom red_subst : ∀O,D,M,N,N1,i. + red O D N N1 → red O D (subst O D M i N) (subst O D M i N1). + +axiom red_star_subst : ∀O,D,M,N,N1,i. + star ? (red O D) N N1 → star ? (red O D) (subst O D M i N) (subst O D M i N1). + +axiom canonical_to_T: ∀O,D.∀M:T O D.∀ty.(* type_of M ty → *) + ∃a:FinSet_of_FType O D ty. star ? (red O D) M (to_T O D ty a). + +axiom normal_to_T: ∀O,D,M,ty,a. red O D (to_T O D ty a) M → False. + +lemma critical: ∀O,D,ty,M,N. + ∃M3:T O D + .star (T O D) (red O D) (subst O D M 0 N) M3 + ∧star (T O D) (red O D) + (App O D + (Vec O D ty + (map (FinSet_of_FType O D ty) (T O D) + (λa0:FinSet_of_FType O D ty.subst O D M 0 (to_T O D ty a0)) + (enum (FinSet_of_FType O D ty)))) N) M3. +#O #D #ty #M #N +lapply (canonical_to_T O D N ty) * #a #Ha +%{(subst O D M 0 (to_T O D ty a))} (* CR-term *) +%[@red_star_subst @Ha + |@trans_star [|@(star_red_appr … Ha)] @R_to_star @riota + lapply (enum_complete (FinSet_of_FType O D ty) a) + elim (enum (FinSet_of_FType O D ty)) + [normalize #H1 destruct (H1) + |#hd #tl #Hind #H cases (orb_true_l … H) -H #Hcase + [normalize >Hcase >(\P Hcase) // + |normalize cases (true_or_false (a==hd)) #Hcase1 + [normalize >Hcase1 >(\P Hcase1) // |>Hcase1 @Hind @Hcase] + ] + ] + ] +qed. + +lemma critical2: ∀O,D,ty,a,M,M1,M2,v. + red O D (Vec O D ty v) M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M1 → + assoc (FinSet_of_FType O D ty) (T O D) a (enum (FinSet_of_FType O D ty)) v + =Some (T O D) M2 → + ∃M3:T O D + .star (T O D) (red O D) M2 M3 + ∧star (T O D) (red O D) (App O D M (to_T O D ty a)) M3. +#O #D #ty #a #M #M1 #M2 #v #redM #redM1 #Ha lapply (red_vec … redM) -redM +* #N * #N1 * #v1 * #v2 * * #Hred1 #Hv #HM0 >HM0 -HM0 >Hv in Ha; #Ha +cases (same_assoc … a (enum (FinSet_of_FType O D ty)) v1 v2 N N1) + [* >Ha -Ha #H1 destruct (H1) #Ha + %{N1} (* CR-term *) % [@R_to_star //|@R_to_star @(riota … Ha)] + |#Ha1 %{M2} (* CR-term *) % [// | @R_to_star @riota HM -HM #HM1 >HM1 - HM1 #Hl inversion Hl + [#P1 #M1 #N1 #H1 destruct (H1) #H_ %{(subst O D M1 0 N1)} (* CR-term *) /2/ + |#ty #v #a #M0 #_ #H1 destruct (H1) (* vacuous *) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #_ inversion redM0 + [#P0 #M0 #N #H destruct + |#ty #v #a #M0 #_ #H1 destruct (H1) + |#M0 #M1 #N #_ #_ #H1 destruct (H1) + |#M0 #M1 #N #_ #_ #H1 destruct (H1) + |#ty1 #M0 #H1 destruct (H1) #HM1 @critical + |#ty #N #N1 #v #v1 #_ #_ #H1 destruct (H1) + ] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #HM2 + %{(subst O D Q 0 N1)} (* CR-term *) + %[@red_star_subst @R_to_star //|@R_to_star @rbeta] + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is iota *)#ty #v #a #M3 #Ha #_ #_ #Hl inversion Hl + [#P1 #M1 #N1 #H1 destruct (H1) (* vacuous *) + |#ty1 #v1 #a1 #M4 #Ha1 #H1 destruct (H1) -H1 #HM4 >(inj_to_T … e0) in Ha; + >Ha1 #H1 destruct (H1) %{M3} (* CR-term *) /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 @(critical2 … redM0 Hl Ha) + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) elim (normal_to_T … redN0N1) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appl *)#M3 #M4 #N #redM3M4 #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M1 #N1 #H1 destruct (H1) #HM2 lapply (red_lambda … redM3M4) + #H >H -H lapply (critical O D ty1 M1 N1) * #M3 * #H1 #H2 + %{M3} /2/ + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct + lapply (critical2 … redM3M4 Hl Ha1) * #M3 * #H1 #H2 %{M3} /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + lapply (HindP … redM0 redM3M4) * #M3 * #H1 #H2 + %{(App O D M3 N0)} (* CR-term *) % [@star_red_appl //|@star_red_appl //] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #_ + %{(App O D M4 N1)} % @R_to_star [@rappr //|@rappl //] + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appr *)#M3 #N #N1 #redN #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M0 #N0 #H1 destruct (H1) #HM2 + %{(subst O D M0 0 N1)} (* CR-term *) % + [@R_to_star @rbeta | @red_star_subst @R_to_star //] + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct (H1) elim (normal_to_T … redN) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + %{(App O D M10 N1)} (* CR-term *) % @R_to_star [@rappl //|@rappr //] + |#M0 #N0 #N10 #redN0 #_ #H1 destruct (H1) #_ + lapply (HindQ … redN0 redN) * #M3 * #H1 #H2 + %{(App O D M0 M3)} (* CR-term *) % [@star_red_appr //|@star_red_appr //] + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is rmem *) #ty #M0 #H1 destruct (H1) (* vacuous *) + |(* right redex is vec *) #ty #N #N1 #v #v1 #_ #_ + #H1 destruct (H1) (* vacuous *) + ] + |#ty #M1 #Hind #M2 #M3 #H1 #H2 + lapply (red_lambda … H1) #HM2 lapply (red_lambda … H2) #HM3 + %{M2} (* CR-term *) % // + |#ty #v1 #Hind #M1 #M2 #H1 #H2 + lapply (red_vec … H1) * #N11 * #N12 * #v11 * #v12 * * #redN11 #Hv1 #HM1 + lapply (red_vec … H2) * #N21* #N22 * #v21 * #v22 * * #redN21 #Hv2 #HM2 + >Hv1 in Hv2; #Hvv lapply (compare_append … Hvv) -Hvv * + (* we must proceed by cases on the list *) * normalize + [(* N11 = N21 *) * + [>append_nil * #Hl1 #Hl2 destruct lapply(Hind N11 … redN11 redN21) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v21@M3::v12))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + |>append_nil * #Hl1 #Hl2 destruct lapply(Hind N21 … redN21 redN11) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v11@M3::v22))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + ] + |(* N11 ≠ N21 *) -Hind #P #l * + [* #Hv11 #Hv22 destruct + %{((Vec O D ty ((v21@N22::l)@N12::v12)))} (* CR-term *) % @R_to_star + [>associative_append >associative_append normalize @rvec // + |>append_cons append_cons associative_append >associative_append normalize @rvec // + ] + ] + ] + ] +qed. + + + + diff --git a/matita/matita/lib/finite_lambda/finite_lambda_deep.ma b/matita/matita/lib/finite_lambda/finite_lambda_deep.ma new file mode 100644 index 000000000..e2613775d --- /dev/null +++ b/matita/matita/lib/finite_lambda/finite_lambda_deep.ma @@ -0,0 +1,562 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "basics/finset.ma". +include "basics/star.ma". + + +inductive FType (O:Type[0]): Type[0] ≝ + | atom : O → FType O + | arrow : FType O → FType O → FType O. + +inductive T (O:Type[0]) (D:O → FinSet): Type[0] ≝ + | Val: ∀o:O.carr (D o) → T O D (* a value in a finset *) + | Rel: nat → T O D (* DB index, base is 0 *) + | App: T O D → T O D → T O D (* function, argument *) + | Lambda: FType O → T O D → T O D (* type, body *) + | Vec: FType O → list (T O D) → T O D (* type, body *) +. + +let rec FinSet_of_FType O (D:O→FinSet) (ty:FType O) on ty : FinSet ≝ + match ty with + [atom o ⇒ D o + |arrow ty1 ty2 ⇒ FinFun (FinSet_of_FType O D ty1) (FinSet_of_FType O D ty2) + ]. + +(* size *) + +let rec size O D (M:T O D) on M ≝ +match M with + [Val o a ⇒ 1 + |Rel n ⇒ 1 + |App P Q ⇒ size O D P + size O D Q + 1 + |Lambda Ty P ⇒ size O D P + 1 + |Vec Ty v ⇒ foldr ?? (λx,a. size O D x + a) 0 v +1 + ] +. + +(* axiom pos_size: ∀M. 1 ≤ size M. *) + +theorem Telim_size: ∀O,D.∀P: T O D → Prop. + (∀M. (∀N. size O D N < size O D M → P N) → P M) → ∀M. P M. +#O #D #P #H #M (cut (∀p,N. size O D N = p → P N)) + [2: /2/] +#p @(nat_elim1 p) #m #H1 #N #sizeN @H #N0 #Hlt @(H1 (size O D N0)) // +qed. + +lemma T_elim: + ∀O: Type[0].∀D:O→FinSet.∀P:T O D→Prop. + (∀o:O.∀x:D o.P (Val O D o x)) → + (∀n:ℕ.P(Rel O D n)) → + (∀m,n:T O D.P m→P n→P (App O D m n)) → + (∀Ty:FType O.∀m:T O D.P m→P(Lambda O D Ty m)) → + (∀Ty:FType O.∀v:list (T O D). + (∀x:T O D. mem ? x v → P x) → P(Vec O D Ty v)) → + ∀x:T O D.P x. +#O #D #P #Hval #Hrel #Happ #Hlam #Hvec @Telim_size #x cases x // + [ (* app *) #m #n #Hind @Happ @Hind // /2 by le_minus_to_plus/ + | (* lam *) #ty #m #Hind @Hlam @Hind normalize // + | (* vec *) #ty #v #Hind @Hvec #x lapply Hind elim v + [#Hind normalize * + |#hd #tl #Hind1 #Hind2 * + [#Hx >Hx @Hind2 normalize // + |@Hind1 #N #H @Hind2 @(lt_to_le_to_lt … H) normalize // + ] + ] + ] +qed. + + +(* arguments: k is the nesting depth (starts from 0), p is the lift *) +let rec lift O D t k p on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then Rel O D (n+p) else Rel O D n + | App m n ⇒ App O D (lift O D m k p) (lift O D n k p) + | Lambda Ty n ⇒ Lambda O D Ty (lift O D n (S k) p) + | Vec Ty v ⇒ Vec O D Ty (map ?? (λx. lift O D x k p) v) + ]. + +notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift 0 $n $M}. +notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}. + +interpretation "Lift" 'Lift n k M = (lift ?? M k n). + +let rec subst O D t k s on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then + (if (eqb k n) then lift O D s 0 n else Rel O D (n-1)) + else(Rel O D n) + | App m n ⇒ App O D (subst O D m k s) (subst O D n k s) + | Lambda T n ⇒ Lambda O D T (subst O D n (S k) s) + | Vec T v ⇒ Vec O D T (map ?? (λx. subst O D x k s) v) + ]. + +(* notation "hvbox(M break [ k ≝ N ])" + non associative with precedence 90 + for @{'Subst1 $M $k $N}. *) + +interpretation "Subst" 'Subst1 M k N = (subst M k N). + +(* closed terms ???? +let rec closed_k O D (t: T O D) k on t ≝ + match t with + [ Val o a ⇒ True + | Rel n ⇒ n < k + | App m n ⇒ (closed_k O D m k) ∧ (closed_k O D n k) + | Lambda T n ⇒ closed_k O D n (k+1) + | Vec T v ⇒ closed_list O D v k + ] + +and closed_list O D (l: list (T O D)) k on l ≝ + match l with + [ nil ⇒ True + | cons hd tl ⇒ closed_k O D hd k ∧ closed_list O D tl k + ] +. *) + +inductive is_closed (O:Type[0]) (D:O→FinSet): nat → T O D → Prop ≝ +| cval : ∀k,o,a.is_closed O D k (Val O D o a) +| cval : ∀k,n. n < k → is_closed O D k (Rel O D n) +| capp : ∀k,n,m. is_closed O D k m → is_closed O D k n → + is_closed O D k (App O D m n) +| clam : ∀T,k,m. is_closed O D (k+1) m → is_closed O D k (Lambda O D T m) +| cvec: ∀T,k,v. (∀m. mem ? m v → is_closed O D k m) → + is_closed O D k (Vec O D T v). + +lemma is_closed_rel: ∀O,D,n,k. + is_closed O D k (Rel O D n) → n < k. +#O #D #n #k #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct // + |#k0 #M #N #_ #_ #_ #H destruct + |#T #k0 #M #_ #_ #H destruct + |#T #k0 #v #_ #_ #H destruct + ] +qed. + + +(*** properties of lift and subst ***) + +lemma lift_0: ∀O,D.∀t:T O D.∀k. lift O D t k 0 = t. +#O #D #t @(T_elim … t) normalize // + [#n #k cases (leb k n) normalize // + |#o #v #Hind #k @eq_f lapply Hind -Hind elim v // + #hd #tl #Hind #Hind1 normalize @eq_f2 + [@Hind1 %1 //|@Hind #x #Hx @Hind1 %2 //] + ] +qed. + +axiom lift_closed: ∀O,D.∀t:T O D.∀k,p. + is_closed O D 0 t → lift O D t k p = t. +(* +#O #D #t @(T_elim … t) normalize // + [#n #k normalize // + |#o #v #Hind #k @eq_f lapply Hind -Hind elim v // + #hd #tl #Hind #Hind1 normalize @eq_f2 + [@Hind1 %1 //|@Hind #x #Hx @Hind1 %2 //] + ] +qed. *) + +let rec to_T O D ty on ty: FinSet_of_FType O D ty → T O D ≝ + match ty return (λty.FinSet_of_FType O D ty → T O D) with + [atom o ⇒ λa.Val O D o a + |arrow ty1 ty2 ⇒ λa:FinFun ??.Vec O D ty1 + (map ((FinSet_of_FType O D ty1)×(FinSet_of_FType O D ty2)) + (T O D) (λp.to_T O D ty2 (snd … p)) (pi1 … a)) + ] +. + +axiom inj_to_T: ∀O,D,ty,a1,a2. to_T O D ty a1 = to_T O D ty a2 → a1 = a2. + +let rec assoc (A:FinSet) (B:Type[0]) (a:A) l1 l2 on l1 : option B ≝ + match l1 with + [ nil ⇒ None ? + | cons hd1 tl1 ⇒ match l2 with + [ nil ⇒ None ? + | cons hd2 tl2 ⇒ if a==hd1 then Some ? hd2 else assoc A B a tl1 tl2 + ] + ]. + +lemma same_assoc: ∀A,B,a,l1,v1,v2,N,N1. + assoc A B a l1 (v1@N::v2) = Some ? N ∧ assoc A B a l1 (v1@N1::v2) = Some ? N1 + ∨ assoc A B a l1 (v1@N::v2) = assoc A B a l1 (v1@N1::v2). +#A #B #a #l1 #v1 #v2 #N #N1 lapply v1 -v1 elim l1 + [#v1 %2 // |#hd #tl #Hind * normalize cases (a==hd) normalize /3/] +qed. + +lemma assoc_to_mem: ∀A,B,a,l1,l2,b. + assoc A B a l1 l2 = Some ? b → mem ? b l2. +#A #B #a #l1 elim l1 + [#l2 #b normalize #H destruct + |#hd1 #tl1 #Hind * + [#b normalize #H destruct + |#hd2 #tl2 #b normalize cases (a==hd1) normalize + [#H %1 destruct //|#H %2 @Hind @H] + ] + ] +qed. + +inductive red (O:Type[0]) (D:O→FinSet) : T O D →T O D → Prop ≝ + | (* we only allow beta on closed arguments *) + rbeta: ∀P,M,N. is_closed O D 0 N → + red O D (App O D (Lambda O D P M) N) (subst O D M 0 N) + | riota: ∀ty,v,a,M. + assoc (FinSet_of_FType O D ty) ? a (enum (FinSet_of_FType O D ty)) v = Some ? M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M + | rappl: ∀M,M1,N. red O D M M1 → red O D (App O D M N) (App O D M1 N) + | rappr: ∀M,N,N1. red O D N N1 → red O D (App O D M N) (App O D M N1) + | rlam: ∀ty,N,N1. red O D N N1 → red O D (Lambda O D ty N) (Lambda O D ty N1) + | rmem: ∀ty,M. red O D (Lambda O D ty M) + (Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty)))) + | rvec: ∀ty,N,N1,v,v1. red O D N N1 → + red O D (Vec O D ty (v@N::v1)) (Vec O D ty (v@N1::v1)). + +(* some inversion cases *) +lemma red_vec: ∀O,D,ty,v,M. + red O D (Vec O D ty v) M → ∃N,N1,v1,v2. + red O D N N1 ∧ v = v1@N::v2 ∧ M = Vec O D ty (v1@N1::v2). +#O #D #ty #v #M #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M #M1 #_ #_ #H destruct + |#ty1 #M0 #H destruct + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct #_ %{N} %{N1} %{v1} %{v2} /3/ + ] +qed. + +lemma red_lambda: ∀O,D,ty,M,N. + red O D (Lambda O D ty M) N → + (∃M1. red O D M M1 ∧ N = (Lambda O D ty M1)) ∨ + N = Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty))). +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #P #P1 #redP #_ #H #H1 destruct %1 %{P1} % // + |#ty1 #M0 #H destruct #_ %2 // + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_val: ∀O,D,ty,a,N. + red O D (Val O D ty a) N → False. +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #N1 #N2 #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_rel: ∀O,D,n,N. + red O D (Rel O D n) N → False. +#O #D #n #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #N1 #N2 #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma star_red_appl: ∀O,D,M,M1,N. star ? (red O D) M M1 → + star ? (red O D) (App O D M N) (App O D M1 N). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappl @HPQ +qed. + +lemma star_red_appr: ∀O,D,M,N,N1. star ? (red O D) N N1 → + star ? (red O D) (App O D M N) (App O D M N1). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappr @HPQ +qed. + +lemma star_red_vec: ∀O,D,ty,N,N1,v1,v2. star ? (red O D) N N1 → + star ? (red O D) (Vec O D ty (v1@N::v2)) (Vec O D ty (v1@N1::v2)). +#O #D #ty #N #N1 #v1 #v2 #H elim H // +#P #Q #Hind #HPQ #Hvec %1[|@Hvec] @rvec @HPQ +qed. + +lemma star_red_vec1: ∀O,D,ty,v1,v2,v. |v1| = |v2| → + (∀n,M. star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) → + star ? (red O D) (Vec O D ty (v@v1)) (Vec O D ty (v@v2)). +#O #D #ty #v1 elim v1 + [#v2 #v normalize #Hv2 >(lenght_to_nil … (sym_eq … Hv2)) normalize // + |#N1 #tl1 #Hind * [normalize #v #H destruct] #N2 #tl2 #v normalize #HS + #H @(trans_star … (Vec O D ty (v@N2::tl1))) + [@star_red_vec @(H 0 N1) + |>append_cons >(append_cons ??? tl2) @(Hind… (injective_S … HS)) + #n #M @(H (S n)) + ] + ] +qed. + +lemma star_red_vec2: ∀O,D,ty,v1,v2. |v1| = |v2| → + (∀n,M. star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) → + star ? (red O D) (Vec O D ty v1) (Vec O D ty v2). +#O #D #ty #v1 #v2 @(star_red_vec1 … [ ]) +qed. + +lemma star_red_lambda: ∀O,D,ty,N,N1. star ? (red O D) N N1 → + star ? (red O D) (Lambda O D ty N) (Lambda O D ty N1). +#O #D #ty #N #N1 #H elim H // +#P #Q #Hind #HPQ #Hlam %1[|@Hlam] @rlam @HPQ +qed. + +axiom red_subst : ∀O,D,M,N,N1,i. + red O D N N1 → red O D (subst O D M i N) (subst O D M i N1). + +axiom red_star_subst : ∀O,D,M,N,N1,i. + star ? (red O D) N N1 → star ? (red O D) (subst O D M i N) (subst O D M i N1). + +axiom red_star_subst2 : ∀O,D,M,M1,N,i. + star ? (red O D) M M1 → star ? (red O D) (subst O D M i N) (subst O D M1 i N). + +axiom canonical_to_T: ∀O,D.∀M:T O D.∀ty.(* type_of M ty → *) + ∃a:FinSet_of_FType O D ty. star ? (red O D) M (to_T O D ty a). + +axiom normal_to_T: ∀O,D,M,ty,a. red O D (to_T O D ty a) M → False. + +axiom red_closed: ∀O,D,M,M1. + is_closed O D 0 M → red O D M M1 → is_closed O D 0 M1. + +lemma critical: ∀O,D,ty,M,N. + ∃M3:T O D + .star (T O D) (red O D) (subst O D M 0 N) M3 + ∧star (T O D) (red O D) + (App O D + (Vec O D ty + (map (FinSet_of_FType O D ty) (T O D) + (λa0:FinSet_of_FType O D ty.subst O D M 0 (to_T O D ty a0)) + (enum (FinSet_of_FType O D ty)))) N) M3. +#O #D #ty #M #N +lapply (canonical_to_T O D N ty) * #a #Ha +%{(subst O D M 0 (to_T O D ty a))} (* CR-term *) +%[@red_star_subst @Ha + |@trans_star [|@(star_red_appr … Ha)] @R_to_star @riota + lapply (enum_complete (FinSet_of_FType O D ty) a) + elim (enum (FinSet_of_FType O D ty)) + [normalize #H1 destruct (H1) + |#hd #tl #Hind #H cases (orb_true_l … H) -H #Hcase + [normalize >Hcase >(\P Hcase) // + |normalize cases (true_or_false (a==hd)) #Hcase1 + [normalize >Hcase1 >(\P Hcase1) // |>Hcase1 @Hind @Hcase] + ] + ] + ] +qed. + +lemma critical2: ∀O,D,ty,a,M,M1,M2,v. + red O D (Vec O D ty v) M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M1 → + assoc (FinSet_of_FType O D ty) (T O D) a (enum (FinSet_of_FType O D ty)) v + =Some (T O D) M2 → + ∃M3:T O D + .star (T O D) (red O D) M2 M3 + ∧star (T O D) (red O D) (App O D M (to_T O D ty a)) M3. +#O #D #ty #a #M #M1 #M2 #v #redM #redM1 #Ha lapply (red_vec … redM) -redM +* #N * #N1 * #v1 * #v2 * * #Hred1 #Hv #HM0 >HM0 -HM0 >Hv in Ha; #Ha +cases (same_assoc … a (enum (FinSet_of_FType O D ty)) v1 v2 N N1) + [* >Ha -Ha #H1 destruct (H1) #Ha + %{N1} (* CR-term *) % [@R_to_star //|@R_to_star @(riota … Ha)] + |#Ha1 %{M2} (* CR-term *) % [// | @R_to_star @riota length_map >length_map //] #n #M0 + cases (true_or_false (leb (|enum (FinSet_of_FType O D ty)|) n)) #Hcase + [>nth_to_default [2:>length_map @(leb_true_to_le … Hcase)] + >nth_to_default [2:>length_map @(leb_true_to_le … Hcase)] // + |cut (n < |enum (FinSet_of_FType O D ty)|) + [@not_le_to_lt @leb_false_to_not_le @Hcase] #Hlt + cut (∃a:FinSet_of_FType O D ty.True) + [lapply Hlt lapply (enum_complete (FinSet_of_FType O D ty)) + cases (enum (FinSet_of_FType O D ty)) + [#_ normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#a #l #_ #_ %{a} // + ] + ] * #a #_ + >(nth_map ?????? a Hlt) >(nth_map ?????? a Hlt) + @red_star_subst2 @R_to_star // + ] + ] +qed. + +(* we need to proceed by structural induction on the term and then +by inversion on the two redexes. The problem are the moves in a +same subterm, since we need an induction hypothesis, there *) + +lemma local_confluence: ∀O,D,M,M1,M2. red O D M M1 → red O D M M2 → +∃M3. star ? (red O D) M1 M3 ∧ star ? (red O D) M2 M3. +#O #D #M @(T_elim … M) + [#o #a #M1 #M2 #H elim(red_val ????? H) + |#n #M1 #M2 #H elim(red_rel ???? H) + |(* app : this is the interesting case *) + #P #Q #HindP #HindQ + #M1 #M2 #H1 inversion H1 -H1 + [(* right redex is beta *) + #ty #Q #N #Hc #HM >HM -HM #HM1 >HM1 - HM1 #Hl inversion Hl + [#ty1 #Q1 #N1 #Hc1 #H1 destruct (H1) #H_ + %{(subst O D Q1 0 N1)} (* CR-term *) /2/ + |#ty #v #a #M0 #_ #H1 destruct (H1) (* vacuous *) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #_ cases (red_lambda … redM0) + [* #Q1 * #redQ #HM10 >HM10 + %{(subst O D Q1 0 N0)} (* CR-term *) % + [@red_star_subst2 @R_to_star //|@R_to_star @rbeta @Hc] + |#HM1 >HM1 @critical + ] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #HM2 + %{(subst O D Q 0 N1)} (* CR-term *) + %[@red_star_subst @R_to_star //|@R_to_star @rbeta @(red_closed … Hc) //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is iota *)#ty #v #a #M3 #Ha #_ #_ #Hl inversion Hl + [#P1 #M1 #N1 #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #v1 #a1 #M4 #Ha1 #H1 destruct (H1) -H1 #HM4 >(inj_to_T … e0) in Ha; + >Ha1 #H1 destruct (H1) %{M3} (* CR-term *) /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 @(critical2 … redM0 Hl Ha) + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) elim (normal_to_T … redN0N1) + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appl *)#M3 #M4 #N #redM3M4 #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M1 #N1 #Hc #H1 destruct (H1) #HM2 lapply (red_lambda … redM3M4) * + [* #M3 * #H1 #H2 >H2 %{(subst O D M3 0 N1)} % + [@R_to_star @rbeta @Hc|@red_star_subst2 @R_to_star @H1] + |#H >H -H lapply (critical O D ty1 M1 N1) * #M3 * #H1 #H2 + %{M3} /2/ + ] + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct + lapply (critical2 … redM3M4 Hl Ha1) * #M3 * #H1 #H2 %{M3} /2/ + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + lapply (HindP … redM0 redM3M4) * #M3 * #H1 #H2 + %{(App O D M3 N0)} (* CR-term *) % [@star_red_appl //|@star_red_appl //] + |#M0 #N0 #N1 #redN0N1 #_ #H1 destruct (H1) #_ + %{(App O D M4 N1)} % @R_to_star [@rappr //|@rappl //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is appr *)#M3 #N #N1 #redN #_ #H1 destruct (H1) #_ + #Hl inversion Hl + [#ty1 #M0 #N0 #Hc #H1 destruct (H1) #HM2 + %{(subst O D M0 0 N1)} (* CR-term *) % + [@R_to_star @rbeta @(red_closed … Hc) //|@red_star_subst @R_to_star // ] + |#ty1 #v1 #a1 #M4 #Ha1 #H1 #H2 destruct (H1) elim (normal_to_T … redN) + |#M0 #M10 #N0 #redM0 #_ #H1 destruct (H1) #HM2 + %{(App O D M10 N1)} (* CR-term *) % @R_to_star [@rappl //|@rappr //] + |#M0 #N0 #N10 #redN0 #_ #H1 destruct (H1) #_ + lapply (HindQ … redN0 redN) * #M3 * #H1 #H2 + %{(App O D M0 M3)} (* CR-term *) % [@star_red_appr //|@star_red_appr //] + |#ty1 #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |#ty1 #M0 #H1 destruct (H1) (* vacuous *) + |#ty1 #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1) (* vacuous *) + ] + |(* right redex is rlam *) #ty #N0 #N1 #_ #_ #H1 destruct (H1) (* vacuous *) + |(* right redex is rmem *) #ty #M0 #H1 destruct (H1) (* vacuous *) + |(* right redex is vec *) #ty #N #N1 #v #v1 #_ #_ + #H1 destruct (H1) (* vacuous *) + ] + |#ty #M1 #Hind #M2 #M3 #H1 #H2 (* this case is not trivial any more *) + lapply (red_lambda … H1) * + [* #M4 * #H3 #H4 >H4 lapply (red_lambda … H2) * + [* #M5 * #H5 #H6 >H6 lapply(Hind … H3 H5) * #M6 * #H7 #H8 + %{(Lambda O D ty M6)} (* CR-term *) % @star_red_lambda // + |#H5 >H5 @critical3 // + ] + |#HM2 >HM2 lapply (red_lambda … H2) * + [* #M4 * #Hred #HM3 >HM3 lapply (critical3 … ty ?? Hred) * #M5 + * #H3 #H4 %{M5} (* CR-term *) % // + |#HM3 >HM3 %{M3} (* CR-term *) % // + ] + ] + |#ty #v1 #Hind #M1 #M2 #H1 #H2 + lapply (red_vec … H1) * #N11 * #N12 * #v11 * #v12 * * #redN11 #Hv1 #HM1 + lapply (red_vec … H2) * #N21* #N22 * #v21 * #v22 * * #redN21 #Hv2 #HM2 + >Hv1 in Hv2; #Hvv lapply (compare_append … Hvv) -Hvv * + (* we must proceed by cases on the list *) * normalize + [(* N11 = N21 *) * + [>append_nil * #Hl1 #Hl2 destruct lapply(Hind N11 … redN11 redN21) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v21@M3::v12))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + |>append_nil * #Hl1 #Hl2 destruct lapply(Hind N21 … redN21 redN11) + [@mem_append_l2 %1 //] + * #M3 * #HM31 #HM32 + %{(Vec O D ty (v11@M3::v22))} (* CR-term *) + % [@star_red_vec //|@star_red_vec //] + ] + |(* N11 ≠ N21 *) -Hind #P #l * + [* #Hv11 #Hv22 destruct + %{((Vec O D ty ((v21@N22::l)@N12::v12)))} (* CR-term *) % @R_to_star + [>associative_append >associative_append normalize @rvec // + |>append_cons append_cons associative_append >associative_append normalize @rvec // + ] + ] + ] + ] +qed. + + + + diff --git a/matita/matita/lib/finite_lambda/reduction.ma b/matita/matita/lib/finite_lambda/reduction.ma new file mode 100644 index 000000000..98c56e10a --- /dev/null +++ b/matita/matita/lib/finite_lambda/reduction.ma @@ -0,0 +1,308 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "finite_lambda/terms_and_types.ma". + +(* some auxiliary lemmas *) + +lemma nth_to_default: ∀A,l,n,d. + |l| ≤ n → nth n A l d = d. +#A #l elim l [//] #a #tl #Hind #n cases n + [#d normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#m #d normalize #H @Hind @le_S_S_to_le @H + ] +qed. + +lemma mem_nth: ∀A,l,n,d. + n < |l| → mem ? (nth n A l d) l. +#A #l elim l + [#n #d normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#a #tl #Hind * normalize + [#_ #_ %1 //| #m #d #HSS %2 @Hind @le_S_S_to_le @HSS] + ] +qed. + +lemma nth_map: ∀A,B,l,f,n,d1,d2. + n < |l| → nth n B (map … f l) d1 = f (nth n A l d2). +#n #B #l #f elim l + [#m #d1 #d2 normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#a #tl #Hind #m #d1 #d2 cases m normalize // + #m1 #H @Hind @le_S_S_to_le @H + ] +qed. + + + +(* end of auxiliary lemmas *) + +let rec to_T O D ty on ty: FinSet_of_FType O D ty → T O D ≝ + match ty return (λty.FinSet_of_FType O D ty → T O D) with + [atom o ⇒ λa.Val O D o a + |arrow ty1 ty2 ⇒ λa:FinFun ??.Vec O D ty1 + (map ((FinSet_of_FType O D ty1)×(FinSet_of_FType O D ty2)) + (T O D) (λp.to_T O D ty2 (snd … p)) (pi1 … a)) + ] +. + +lemma is_closed_to_T: ∀O,D,ty,a. is_closed O D 0 (to_T O D ty a). +#O #D #ty elim ty // +#ty1 #ty2 #Hind1 #Hind2 #a normalize @cvec #m #Hmem +lapply (mem_map ????? Hmem) * #a1 * #H1 #H2

H //] -Hl1 -Hl2 + lapply e0 -e0 lapply l2 -l2 elim l1 + [#l2 cases l2 normalize [// |#a1 #tl1 #H destruct] + |#a1 #tl1 #Hind #l2 cases l2 + [normalize #H destruct + |#a2 #tl2 normalize #H @eq_f2 + [@Hind2 *) + +let rec assoc (A:FinSet) (B:Type[0]) (a:A) l1 l2 on l1 : option B ≝ + match l1 with + [ nil ⇒ None ? + | cons hd1 tl1 ⇒ match l2 with + [ nil ⇒ None ? + | cons hd2 tl2 ⇒ if a==hd1 then Some ? hd2 else assoc A B a tl1 tl2 + ] + ]. + +lemma same_assoc: ∀A,B,a,l1,v1,v2,N,N1. + assoc A B a l1 (v1@N::v2) = Some ? N ∧ assoc A B a l1 (v1@N1::v2) = Some ? N1 + ∨ assoc A B a l1 (v1@N::v2) = assoc A B a l1 (v1@N1::v2). +#A #B #a #l1 #v1 #v2 #N #N1 lapply v1 -v1 elim l1 + [#v1 %2 // |#hd #tl #Hind * normalize cases (a==hd) normalize /3/] +qed. + +lemma assoc_to_mem: ∀A,B,a,l1,l2,b. + assoc A B a l1 l2 = Some ? b → mem ? b l2. +#A #B #a #l1 elim l1 + [#l2 #b normalize #H destruct + |#hd1 #tl1 #Hind * + [#b normalize #H destruct + |#hd2 #tl2 #b normalize cases (a==hd1) normalize + [#H %1 destruct //|#H %2 @Hind @H] + ] + ] +qed. + +lemma assoc_to_mem2: ∀A,B,a,l1,l2,b. + assoc A B a l1 l2 = Some ? b → ∃l21,l22.l2=l21@b::l22. +#A #B #a #l1 elim l1 + [#l2 #b normalize #H destruct + |#hd1 #tl1 #Hind * + [#b normalize #H destruct + |#hd2 #tl2 #b normalize cases (a==hd1) normalize + [#H %{[]} %{tl2} destruct // + |#H lapply (Hind … H) * #la * #lb #H1 + %{(hd2::la)} %{lb} >H1 //] + ] + ] +qed. + +lemma assoc_map: ∀A,B,C,a,l1,l2,f,b. + assoc A B a l1 l2 = Some ? b → assoc A C a l1 (map ?? f l2) = Some ? (f b). +#A #B #C #a #l1 elim l1 + [#l2 #f #b normalize #H destruct + |#hd1 #tl1 #Hind * + [#f #b normalize #H destruct + |#hd2 #tl2 #f #b normalize cases (a==hd1) normalize + [#H destruct // |#H @(Hind … H)] + ] + ] +qed. + +(*************************** One step reduction *******************************) + +inductive red (O:Type[0]) (D:O→FinSet) : T O D →T O D → Prop ≝ + | (* we only allow beta on closed arguments *) + rbeta: ∀P,M,N. is_closed O D 0 N → + red O D (App O D (Lambda O D P M) N) (subst O D M 0 N) + | riota: ∀ty,v,a,M. + assoc ?? a (enum (FinSet_of_FType O D ty)) v = Some ? M → + red O D (App O D (Vec O D ty v) (to_T O D ty a)) M + | rappl: ∀M,M1,N. red O D M M1 → red O D (App O D M N) (App O D M1 N) + | rappr: ∀M,N,N1. red O D N N1 → red O D (App O D M N) (App O D M N1) + | rlam: ∀ty,N,N1. red O D N N1 → red O D (Lambda O D ty N) (Lambda O D ty N1) + | rmem: ∀ty,M. red O D (Lambda O D ty M) + (Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty)))) + | rvec: ∀ty,N,N1,v,v1. red O D N N1 → + red O D (Vec O D ty (v@N::v1)) (Vec O D ty (v@N1::v1)). + +(*********************************** inversion ********************************) +lemma red_vec: ∀O,D,ty,v,M. + red O D (Vec O D ty v) M → ∃N,N1,v1,v2. + red O D N N1 ∧ v = v1@N::v2 ∧ M = Vec O D ty (v1@N1::v2). +#O #D #ty #v #M #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #M #M1 #_ #_ #H destruct + |#ty1 #M0 #H destruct + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct #_ %{N} %{N1} %{v1} %{v2} /3/ + ] +qed. + +lemma red_lambda: ∀O,D,ty,M,N. + red O D (Lambda O D ty M) N → + (∃M1. red O D M M1 ∧ N = (Lambda O D ty M1)) ∨ + N = Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a)) + (enum (FinSet_of_FType O D ty))). +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #P #P1 #redP #_ #H #H1 destruct %1 %{P1} % // + |#ty1 #M0 #H destruct #_ %2 // + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_val: ∀O,D,ty,a,N. + red O D (Val O D ty a) N → False. +#O #D #ty #M #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #N1 #N2 #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +lemma red_rel: ∀O,D,n,N. + red O D (Rel O D n) N → False. +#O #D #n #N #Hred inversion Hred + [#ty1 #M0 #N #Hc #H destruct + |#ty1 #v1 #a #M0 #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#M0 #M1 #N #_ #_ #H destruct + |#ty1 #N1 #N2 #_ #_ #H destruct + |#ty1 #M0 #H destruct #_ + |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct + ] +qed. + +(*************************** multi step reduction *****************************) +lemma star_red_appl: ∀O,D,M,M1,N. star ? (red O D) M M1 → + star ? (red O D) (App O D M N) (App O D M1 N). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappl @HPQ +qed. + +lemma star_red_appr: ∀O,D,M,N,N1. star ? (red O D) N N1 → + star ? (red O D) (App O D M N) (App O D M N1). +#O #D #M #N #N1 #H elim H // +#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappr @HPQ +qed. + +lemma star_red_vec: ∀O,D,ty,N,N1,v1,v2. star ? (red O D) N N1 → + star ? (red O D) (Vec O D ty (v1@N::v2)) (Vec O D ty (v1@N1::v2)). +#O #D #ty #N #N1 #v1 #v2 #H elim H // +#P #Q #Hind #HPQ #Hvec %1[|@Hvec] @rvec @HPQ +qed. + +lemma star_red_vec1: ∀O,D,ty,v1,v2,v. |v1| = |v2| → + (∀n,M. n < |v1| → star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) → + star ? (red O D) (Vec O D ty (v@v1)) (Vec O D ty (v@v2)). +#O #D #ty #v1 elim v1 + [#v2 #v normalize #Hv2 >(lenght_to_nil … (sym_eq … Hv2)) normalize // + |#N1 #tl1 #Hind * [normalize #v #H destruct] #N2 #tl2 #v normalize #HS + #H @(trans_star … (Vec O D ty (v@N2::tl1))) + [@star_red_vec @(H 0 N1) @le_S_S // + |>append_cons >(append_cons ??? tl2) @(Hind… (injective_S … HS)) + #n #M #H1 @(H (S n)) @le_S_S @H1 + ] + ] +qed. + +lemma star_red_vec2: ∀O,D,ty,v1,v2. |v1| = |v2| → + (∀n,M. n < |v1| → star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) → + star ? (red O D) (Vec O D ty v1) (Vec O D ty v2). +#O #D #ty #v1 #v2 @(star_red_vec1 … [ ]) +qed. + +lemma star_red_lambda: ∀O,D,ty,N,N1. star ? (red O D) N N1 → + star ? (red O D) (Lambda O D ty N) (Lambda O D ty N1). +#O #D #ty #N #N1 #H elim H // +#P #Q #Hind #HPQ #Hlam %1[|@Hlam] @rlam @HPQ +qed. + +(************************ reduction and substitution **************************) + +lemma red_star_subst : ∀O,D,M,N,N1,i. + star ? (red O D) N N1 → star ? (red O D) (subst O D M i N) (subst O D M i N1). +#O #D #M #N #N1 #i #Hred lapply i -i @(T_elim … M) normalize + [#o #a #i // + |#i #n cases (leb n i) normalize // cases (eqb n i) normalize // + |#P #Q #HindP #HindQ #n normalize + @(trans_star … (App O D (subst O D P n N1) (subst O D Q n N))) + [@star_red_appl @HindP |@star_red_appr @HindQ] + |#ty #P #HindP #i @star_red_lambda @HindP + |#ty #v #Hindv #i @star_red_vec2 [>length_map >length_map //] + #j #Q inversion v [#_ normalize //] #a #tl #_ #Hv + cases (true_or_false (leb (S j) (|a::tl|))) #Hcase + [lapply (leb_true_to_le … Hcase) -Hcase #Hcase + >(nth_map ?????? a Hcase) >(nth_map ?????? a Hcase) #_ @Hindv >Hv @mem_nth // + |>nth_to_default + [2:>length_map @le_S_S_to_le @not_le_to_lt @leb_false_to_not_le //] + >nth_to_default + [2:>length_map @le_S_S_to_le @not_le_to_lt @leb_false_to_not_le //] // + ] + ] +qed. + +lemma red_star_subst2 : ∀O,D,M,M1,N,i. is_closed O D 0 N → + red O D M M1 → star ? (red O D) (subst O D M i N) (subst O D M1 i N). +#O #D #M #M1 #N #i #HNc #Hred lapply i -i elim Hred + [#ty #P #Q #HQc #i normalize @starl_to_star @sstepl + [|@rbeta >(subst_closed … HQc) //] >(subst_closed … HQc) // + lapply (subst_lemma ?? P ?? i 0 (is_closed_mono … HQc) HNc) // + (subst_closed … (le_O_n …)) // + @R_to_star @riota @assoc_map @HP + |#P #P1 #Q #Hred #Hind #i normalize @star_red_appl @Hind + |#P #P1 #Q #Hred #Hind #i normalize @star_red_appr @Hind + |#ty #P #P1 #Hred #Hind #i normalize @star_red_lambda @Hind + |#ty #P #i normalize @starl_to_star @sstepl [|@rmem] + @star_to_starl @star_red_vec2 [>length_map >length_map >length_map //] + #n #Q >length_map #H + cut (∃a:(FinSet_of_FType O D ty).True) + [lapply H -H lapply (enum_complete (FinSet_of_FType O D ty)) + cases (enum (FinSet_of_FType O D ty)) + [#x normalize #H @False_ind @(absurd … H) @lt_to_not_le // + |#x #l #_ #_ %{x} // + ] + ] * #a #_ + >(nth_map ?????? a H) >(nth_map ?????? Q) [2:>length_map @H] + >(nth_map ?????? a H) + lapply (subst_lemma O D P (to_T O D ty + (nth n (FinSet_of_FType O D ty) (enum (FinSet_of_FType O D ty)) a)) + N i 0 (is_closed_mono … (is_closed_to_T …)) HNc) // H1 + Hx @Hind2 normalize // + |@Hind1 #N #H @Hind2 @(lt_to_le_to_lt … H) normalize // + ] + ] + ] +qed. + +(* since we only consider beta reduction with closed arguments we could avoid +lifting. We define it for the sake of generality *) + +(* arguments: k is the nesting depth (starts from 0), p is the lift +let rec lift O D t k p on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then Rel O D (n+p) else Rel O D n + | App m n ⇒ App O D (lift O D m k p) (lift O D n k p) + | Lambda Ty n ⇒ Lambda O D Ty (lift O D n (S k) p) + | Vec Ty v ⇒ Vec O D Ty (map ?? (λx. lift O D x k p) v) + ]. + +notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift 0 $n $M}. +notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}. + +interpretation "Lift" 'Lift n k M = (lift ?? M k n). + +let rec subst O D t k s on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then + (if (eqb k n) then lift O D s 0 n else Rel O D (n-1)) + else(Rel O D n) + | App m n ⇒ App O D (subst O D m k s) (subst O D n k s) + | Lambda T n ⇒ Lambda O D T (subst O D n (S k) s) + | Vec T v ⇒ Vec O D T (map ?? (λx. subst O D x k s) v) + ]. +*) + +(* simplified version of subst, assuming the argument s is closed *) + +let rec subst O D t k s on t ≝ + match t with + [ Val o a ⇒ Val O D o a + | Rel n ⇒ if (leb k n) then + (if (eqb k n) then (* lift O D s 0 n*) s else Rel O D (n-1)) + else(Rel O D n) + | App m n ⇒ App O D (subst O D m k s) (subst O D n k s) + | Lambda T n ⇒ Lambda O D T (subst O D n (S k) s) + | Vec T v ⇒ Vec O D T (map ?? (λx. subst O D x k s) v) + ]. +(* notation "hvbox(M break [ k ≝ N ])" + non associative with precedence 90 + for @{'Subst1 $M $k $N}. *) + +interpretation "Subst" 'Subst1 M k N = (subst M k N). + +(* +lemma subst_rel1: ∀O,D,A.∀k,i. i < k → + (Rel O D i) [k ≝ A] = Rel O D i. +#A #k #i normalize #ltik >(lt_to_leb_false … ltik) // +qed. + +lemma subst_rel2: ∀O,D, A.∀k. + (Rel k) [k ≝ A] = lift A 0 k. +#A #k normalize >(le_to_leb_true k k) // >(eq_to_eqb_true … (refl …)) // +qed. + +lemma subst_rel3: ∀A.∀k,i. k < i → + (Rel i) [k ≝ A] = Rel (i-1). +#A #k #i normalize #ltik >(le_to_leb_true k i) /2/ +>(not_eq_to_eqb_false k i) // @lt_to_not_eq // +qed. *) + + +(* closed terms ???? +let rec closed_k O D (t: T O D) k on t ≝ + match t with + [ Val o a ⇒ True + | Rel n ⇒ n < k + | App m n ⇒ (closed_k O D m k) ∧ (closed_k O D n k) + | Lambda T n ⇒ closed_k O D n (k+1) + | Vec T v ⇒ closed_list O D v k + ] + +and closed_list O D (l: list (T O D)) k on l ≝ + match l with + [ nil ⇒ True + | cons hd tl ⇒ closed_k O D hd k ∧ closed_list O D tl k + ] +. *) + +inductive is_closed (O:Type[0]) (D:O→FinSet): nat → T O D → Prop ≝ +| cval : ∀k,o,a.is_closed O D k (Val O D o a) +| crel : ∀k,n. n < k → is_closed O D k (Rel O D n) +| capp : ∀k,m,n. is_closed O D k m → is_closed O D k n → + is_closed O D k (App O D m n) +| clam : ∀T,k,m. is_closed O D (S k) m → is_closed O D k (Lambda O D T m) +| cvec: ∀T,k,v. (∀m. mem ? m v → is_closed O D k m) → + is_closed O D k (Vec O D T v). + +lemma is_closed_rel: ∀O,D,n,k. + is_closed O D k (Rel O D n) → n < k. +#O #D #n #k #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct // + |#k0 #M #N #_ #_ #_ #_ #_ #H destruct + |#T #k0 #M #_ #_ #_ #H destruct + |#T #k0 #v #_ #_ #_ #H destruct + ] +qed. + +lemma is_closed_app: ∀O,D,k,M, N. + is_closed O D k (App O D M N) → is_closed O D k M ∧ is_closed O D k N. +#O #D #k #M #N #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct + |#k0 #M1 #N1 #HM #HN #_ #_ #_ #H1 destruct % // + |#T #k0 #M #_ #_ #_ #H destruct + |#T #k0 #v #_ #_ #_ #H destruct + ] +qed. + +lemma is_closed_lam: ∀O,D,k,ty,M. + is_closed O D k (Lambda O D ty M) → is_closed O D (S k) M. +#O #D #k #ty #M #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct + |#k0 #M1 #N1 #HM #HN #_ #_ #_ #H1 destruct + |#T #k0 #M1 #HM1 #_ #_ #H1 destruct // + |#T #k0 #v #_ #_ #_ #H destruct + ] +qed. + +lemma is_closed_vec: ∀O,D,k,ty,v. + is_closed O D k (Vec O D ty v) → ∀m. mem ? m v → is_closed O D k m. +#O #D #k #ty #M #H inversion H + [#k0 #o #a #eqk #H destruct + |#k0 #n0 #ltn0 #eqk #H destruct + |#k0 #M1 #N1 #HM #HN #_ #_ #_ #H1 destruct + |#T #k0 #M1 #HM1 #_ #_ #H1 destruct + |#T #k0 #v #Hv #_ #_ #H1 destruct @Hv + ] +qed. + +lemma is_closed_S: ∀O,D,M,m. + is_closed O D m M → is_closed O D (S m) M. +#O #D #M #m #H elim H // + [#k #n0 #Hlt @crel @le_S // + |#k #P #Q #HP #HC #H1 #H2 @capp // + |#ty #k #P #HP #H1 @clam // + |#ty #k #v #Hind #Hv @cvec @Hv + ] +qed. + +lemma is_closed_mono: ∀O,D,M,m,n. m ≤ n → + is_closed O D m M → is_closed O D n M. +#O #D #M #m #n #lemn elim lemn // #i #j #H #H1 @is_closed_S @H @H1 +qed. + + +(*** properties of lift and subst ***) + +(* +lemma lift_0: ∀O,D.∀t:T O D.∀k. lift O D t k 0 = t. +#O #D #t @(T_elim … t) normalize // + [#n #k cases (leb k n) normalize // + |#o #v #Hind #k @eq_f lapply Hind -Hind elim v // + #hd #tl #Hind #Hind1 normalize @eq_f2 + [@Hind1 %1 //|@Hind #x #Hx @Hind1 %2 //] + ] +qed. + +lemma lift_closed: ∀O,D.∀t:T O D.∀k,p. + is_closed O D k t → lift O D t k p = t. +#O #D #t @(T_elim … t) normalize // + [#n #k #p #H >(not_le_to_leb_false … (lt_to_not_le … (is_closed_rel … H))) // + |#M #N #HindM #HindN #k #p #H lapply (is_closed_app … H) * #HcM #HcN + >(HindM … HcM) >(HindN … HcN) // + |#ty #M #HindM #k #p #H lapply (is_closed_lam … H) -H #H >(HindM … H) // + |#ty #v #HindM #k #p #H lapply (is_closed_vec … H) -H #H @eq_f + cut (∀m. mem ? m v → lift O D m k p = m) + [#m #Hmem @HindM [@Hmem | @H @Hmem]] -HindM + elim v // #a #tl #Hind #H1 normalize @eq_f2 + [@H1 %1 //|@Hind #m #Hmem @H1 %2 @Hmem] + ] +qed. + +*) + +lemma subst_closed: ∀O,D,M,N,k,i. k ≤ i → + is_closed O D k M → subst O D M i N = M. +#O #D #M @(T_elim … M) + [#o #a normalize // + |#n #N #k #j #Hlt #Hc lapply (is_closed_rel … Hc) #Hnk normalize + >not_le_to_leb_false [2:@lt_to_not_le @(lt_to_le_to_lt … Hnk Hlt)] // + |#P #Q #HindP #HindQ #N #k #i #ltki #Hc lapply (is_closed_app … Hc) * + #HcP #HcQ normalize >(HindP … ltki HcP) >(HindQ … ltki HcQ) // + |#ty #P #HindP #N #k #i #ltki #Hc lapply (is_closed_lam … Hc) + #HcP normalize >(HindP … HcP) // @le_S_S @ltki + |#ty #v #Hindv #N #k #i #ltki #Hc lapply (is_closed_vec … Hc) + #Hcv normalize @eq_f + cut (∀m:T O D.mem (T O D) m v→ subst O D m i N=m) + [#m #Hmem @(Hindv … Hmem N … ltki) @Hcv @Hmem] + elim v // #a #tl #Hind #H normalize @eq_f2 + [@H %1 //| @Hind #Hmem #Htl @H %2 @Htl] + ] +qed. + +lemma subst_lemma: ∀O,D,A,B,C,k,i. is_closed O D k B → is_closed O D i C → + subst O D (subst O D A i B) (k+i) C = + subst O D (subst O D A (k+S i) C) i B. +#O #D #A #B #C #k @(T_elim … A) normalize + [// + |#n #i #HBc #HCc @(leb_elim i n) #Hle + [@(eqb_elim i n) #eqni + [(lt_to_leb_false (k+(S i)) i) // normalize + >(subst_closed … HBc) // >le_to_leb_true // >eq_to_eqb_true // + |(cut (i < n)) + [cases (le_to_or_lt_eq … Hle) // #eqin @False_ind /2/] #ltin + (cut (0 < n)) [@(le_to_lt_to_lt … ltin) //] #posn + normalize @(leb_elim (k+i) (n-1)) #nk + [@(eqb_elim (k+i) (n-1)) #H normalize + [cut (k+(S i) = n); [/2 by S_pred/] #H1 + >(le_to_leb_true (k+(S i)) n) /2/ + >(eq_to_eqb_true … H1) normalize >(subst_closed … HCc) // + |(cut (k+i < n-1)) [@not_eq_to_le_to_lt; //] #Hlt + >(le_to_leb_true (k+(S i)) n) normalize + [>(not_eq_to_eqb_false (k+(S i)) n) normalize + [>le_to_leb_true [2:@lt_to_le @(le_to_lt_to_lt … Hlt) //] + >not_eq_to_eqb_false // @lt_to_not_eq @(le_to_lt_to_lt … Hlt) // + |@(not_to_not … H) #Hn /2 by plus_minus/ + ] + |(not_le_to_leb_false (k+(S i)) n) normalize + [>(le_to_leb_true … Hle) >(not_eq_to_eqb_false … eqni) // + |@(not_to_not … nk) #H @le_plus_to_minus_r // + ] + ] + ] + |(cut (n < k+i)) [@(lt_to_le_to_lt ? i) /2 by not_le_to_lt/] #ltn + >not_le_to_leb_false [2: @lt_to_not_le @(transitive_lt …ltn) //] normalize + >not_le_to_leb_false [2: @lt_to_not_le //] normalize + >(not_le_to_leb_false … Hle) // + ] + |#M #N #HindM #HindN #i #HBC #HCc @eq_f2 [@HindM // |@HindN //] + |#ty #M #HindM #i #HBC #HCc @eq_f >plus_n_Sm >plus_n_Sm @HindM // + @is_closed_S // + |#ty #v #Hindv #i #HBC #HCc @eq_f + cut (∀m.mem ? m v → subst O D (subst O D m i B) (k+i) C = + subst O D (subst O D m (k+S i) C) i B) + [#m #Hmem @Hindv //] -Hindv elim v normalize [//] + #a #tl #Hind #H @eq_f2 [@H %1 // | @Hind #m #Hmem @H %2 //] + ] +qed. + + diff --git a/matita/matita/lib/finite_lambda/typing.ma b/matita/matita/lib/finite_lambda/typing.ma new file mode 100644 index 000000000..791e27d66 --- /dev/null +++ b/matita/matita/lib/finite_lambda/typing.ma @@ -0,0 +1,229 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "finite_lambda/reduction.ma". + + +(****************************************************************) + +inductive TJ (O: Type[0]) (D:O → FinSet): list (FType O) → T O D → FType O → Prop ≝ + | tval: ∀G,o,a. TJ O D G (Val O D o a) (atom O o) + | trel: ∀G1,ty,G2,n. length ? G1 = n → TJ O D (G1@ty::G2) (Rel O D n) ty + | tapp: ∀G,M,N,ty1,ty2. TJ O D G M (arrow O ty1 ty2) → TJ O D G N ty1 → + TJ O D G (App O D M N) ty2 + | tlambda: ∀G,M,ty1,ty2. TJ O D (ty1::G) M ty2 → + TJ O D G (Lambda O D ty1 M) (arrow O ty1 ty2) + | tvec: ∀G,v,ty1,ty2. + (|v| = |enum (FinSet_of_FType O D ty1)|) → + (∀M. mem ? M v → TJ O D G M ty2) → + TJ O D G (Vec O D ty1 v) (arrow O ty1 ty2). + +lemma wt_to_T: ∀O,D,G,ty,a.TJ O D G (to_T O D ty a) ty. +#O #D #G #ty elim ty + [#o #a normalize @tval + |#ty1 #ty2 #Hind1 #Hind2 normalize * #v #Hv @tvec + [length_map >length_map // + |#M elim v + [normalize @False_ind |#a #v1 #Hind3 * [#eqM >eqM @Hind2 |@Hind3]] + ] + ] +qed. + +lemma inv_rel: ∀O,D,G,n,ty. + TJ O D G (Rel O D n) ty → ∃G1,G2.|G1|=n∧G=G1@ty::G2. +#O #D #G #n #ty #Hrel inversion Hrel + [#G1 #o #a #_ #H destruct + |#G1 #ty1 #G2 #n1 #H1 #H2 #H3 #H4 destruct %{G1} %{G2} /2/ + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tlambda: ∀O,D,G,M,ty1,ty2,ty3. + TJ O D G (Lambda O D ty1 M) (arrow O ty2 ty3) → + ty1 = ty2 ∧ TJ O D (ty2::G) M ty3. +#O #D #G #M #ty1 #ty2 #ty3 #Hlam inversion Hlam + [#G1 #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct % // + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tvec: ∀O,D,G,v,ty1,ty2,ty3. + TJ O D G (Vec O D ty1 v) (arrow O ty2 ty3) → + (|v| = |enum (FinSet_of_FType O D ty1)|) ∧ + (∀M. mem ? M v → TJ O D G M ty3). +#O #D #G #v #ty1 #ty2 #ty3 #Hvec inversion Hvec + [#G #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v1 #ty4 #ty5 #Hv #Hmem #_ #_ #H #H1 destruct % // @Hmem + ] +qed. + +(* could be generalized *) +lemma weak_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@G2) (Rel O D n) ty1 → + TJ O D (G1@ty2::G2) (Rel O D (S n)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |* #H3 #H4 >H4 >append_cons length_append >length_append

H3 >length_append normalize + >plus_n_Sm >associative_plus @eq_f // + ] +qed. + +lemma strength_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@ty2::G2) (Rel O D n) ty1 → + TJ O D (G1@G2) (Rel O D (n-1)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |lapply G5 -G5 * + [>append_nil normalize * #H3 #H4 destruct @False_ind @(absurd … HG1) + @le_to_not_lt // + |#ty3 #G5 * #H3 normalize #H4 destruct (H4) H3 >length_append >length_append normalize H1 >length_append // + |* cases G6 + [>append_nil normalize #H1 @False_ind + @(absurd ? Hlt) @le_to_not_lt H1 // + |#ty1 #G7 #H1 normalize #H2 destruct >associative_append @trel // + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #G3 + #Heq #Hc lapply (is_closed_app … Hc) -Hc * #HMc #HNc + @(tapp … (HindM … Heq HMc) (HindN … Heq HNc)) + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #G3 #Heq #Hc + lapply (is_closed_lam … Hc) -Hc #HMc + @tlambda @(HindM (ty1::G1) G2) [>Heq // |@HMc] + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #G3 #H1 #Hc @tvec + [>length_map // + |#M #Hmem @Hind // lapply (is_closed_vec … Hc) #Hvc @Hvc // + ] + ] +qed. + +lemma nth_spec: ∀A,a,d,l1,l2,n. |l1| = n → nth n A (l1@a::l2) d = a. +#A #a #d #l1 elim l1 normalize + [#l2 #n #Hn H cases (le_to_or_lt_eq … (leb_true_to_le … H)) + [#ltn >(not_eq_to_eqb_false … (lt_to_not_eq … ltn)) normalize + lapply (compare_append … HG) * #G3 * + [* #HG1 #HG2 @(strength_rel … tyN … ltn) HG in ltn; >length_append #ltn @False_ind + @(absurd … ltn) @le_to_not_lt >Hlen // + ] + |#HG21 >(eq_to_eqb_true … HG21) + cut (ty = tyN) + [<(nth_spec ? ty ty ? G2 … Hlen) >HG @nth_spec @HG21] #Hty >Hty + normalize H normalize lapply (compare_append … HG) * #G3 * + [* #H1 @False_ind @(absurd ? Hlen) @sym_not_eq @lt_to_not_eq >H1 + >length_append @(lt_to_le_to_lt n (|G21|)) // @not_le_to_lt + @(leb_false_to_not_le … H) + |cases G3 + [>append_nil * #H1 @False_ind @(absurd ? Hlen)

associative_append @trel // + ] + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #N0 #tyN0 #eqG + #HN0 #Hc normalize @(tapp … ty1) + [@(HindM … eqG HN0 Hc) |@(HindN … eqG HN0 Hc)] + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #N0 #tyN0 #eqG + #HN0 #Hc normalize @(tlambda … ty1) @(HindM (ty1::G1) … HN0) // >eqG // + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #N0 #tyN0 #eqG + #HN0 #Hc normalize @(tvec … ty1) + [>length_map @Hlen + |#M #Hmem lapply (mem_map ????? Hmem) * #a * -Hmem #Hmem #eqM HM1 @tlambda @Hind // + |#HM1 >HM1 @tvec // #N #HN lapply(mem_map ????? HN) + * #a * #mema #eqN H3 @tvec + [H2 >length_append >length_append @eq_f // + |#M2 #Hmem cases (mem_append ???? Hmem) -Hmem #Hmem + [@Hv >H2 @mem_append_l1 // + |cases Hmem + [#HM2 >HM2 -HM2 @(Hind N … H1) >H2 @mem_append_l2 %1 // + |-Hmem #Hmem @Hv >H2 @mem_append_l2 %2 // + ] + ] + ] + ] +qed. + diff --git a/matita/matita/lib/finite_lambda/typing_deep.ma b/matita/matita/lib/finite_lambda/typing_deep.ma new file mode 100644 index 000000000..d2fde8760 --- /dev/null +++ b/matita/matita/lib/finite_lambda/typing_deep.ma @@ -0,0 +1,239 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "finite_lambda/finite_lambda_deep.ma". + + +(****************************************************************) + +inductive TJ (O: Type[0]) (D:O → FinSet): list (FType O) → T O D → FType O → Prop ≝ + | tval: ∀G,o,a. TJ O D G (Val O D o a) (atom O o) + | trel: ∀G1,ty,G2,n. length ? G1 = n → TJ O D (G1@ty::G2) (Rel O D n) ty + | tapp: ∀G,M,N,ty1,ty2. TJ O D G M (arrow O ty1 ty2) → TJ O D G N ty1 → + TJ O D G (App O D M N) ty2 + | tlambda: ∀G,M,ty1,ty2. TJ O D (ty1::G) M ty2 → + TJ O D G (Lambda O D ty1 M) (arrow O ty1 ty2) + | tvec: ∀G,v,ty1,ty2. + (|v| = |enum (FinSet_of_FType O D ty1)|) → + (∀M. mem ? M v → TJ O D G M ty2) → + TJ O D G (Vec O D ty1 v) (arrow O ty1 ty2). + +lemma wt_to_T: ∀O,D,G,ty,a.TJ O D G (to_T O D ty a) ty. +#O #D #G #ty elim ty + [#o #a normalize @tval + |#ty1 #ty2 #Hind1 #Hind2 normalize * #v #Hv @tvec + [length_map >length_map // + |#M elim v + [normalize @False_ind |#a #v1 #Hind3 * [#eqM >eqM @Hind2 |@Hind3]] + ] + ] +qed. + +lemma inv_rel: ∀O,D,G,n,ty. + TJ O D G (Rel O D n) ty → ∃G1,G2.|G1|=n∧G=G1@ty::G2. +#O #D #G #n #ty #Hrel inversion Hrel + [#G1 #o #a #_ #H destruct + |#G1 #ty1 #G2 #n1 #H1 #H2 #H3 #H4 destruct %{G1} %{G2} /2/ + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tlambda: ∀O,D,G,M,ty1,ty2,ty3. + TJ O D G (Lambda O D ty1 M) (arrow O ty2 ty3) → + ty1 = ty2 ∧ TJ O D (ty2::G) M ty3. +#O #D #G #M #ty1 #ty2 #ty3 #Hlam inversion Hlam + [#G1 #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct % // + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tvec: ∀O,D,G,v,ty1,ty2,ty3. + TJ O D G (Vec O D ty1 v) (arrow O ty2 ty3) → + (|v| = |enum (FinSet_of_FType O D ty1)|) ∧ + (∀M. mem ? M v → TJ O D G M ty3). +#O #D #G #v #ty1 #ty2 #ty3 #Hvec inversion Hvec + [#G #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v1 #ty4 #ty5 #Hv #Hmem #_ #_ #H #H1 destruct % // @Hmem + ] +qed. + +(* could be generalized *) +lemma weak_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@G2) (Rel O D n) ty1 → + TJ O D (G1@ty2::G2) (Rel O D (S n)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |* #H3 #H4 >H4 >append_cons length_append >length_append

H3 >length_append normalize + >plus_n_Sm >associative_plus @eq_f // + ] +qed. + +lemma strength_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@ty2::G2) (Rel O D n) ty1 → + TJ O D (G1@G2) (Rel O D (n-1)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |lapply G5 -G5 * + [>append_nil normalize * #H3 #H4 destruct @False_ind @(absurd … HG1) + @le_to_not_lt // + |#ty3 #G5 * #H3 normalize #H4 destruct (H4) H3 >length_append >length_append normalize Hcase normalize + [lapply (compare_append … H) * #G6 * + [* #H1 #H2 >H2 H1 >length_append >length_append >length_append // + |cases G6 + [* >append_nil normalize #H1 #H2

H1 >length_append // + |#ty1 #G7 * #H @False_ind @(absurd … (leb_true_to_le … Hcase)) + @lt_to_not_le H >length_append normalize // + ] + ] + |lapply (compare_append … H) * #G6 * + [* #H1 @False_ind @(absurd ?? (leb_false_to_not_le … Hcase)) H1 + >length_append normalize // + |* cases G6 + [>append_nil normalize #H1 @False_ind + @(absurd ?? (leb_false_to_not_le … Hcase)) H1 // + |#ty1 #G7 #H1 normalize #H2 destruct >associative_append @trel // + ] + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #G3 + #Heq @(tapp … (HindM … Heq) (HindN … Heq)) + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #G3 #Heq @tlambda @(HindM (ty1::G1)) + >Heq // + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #G3 #H1 @tvec + [>length_map // + |#M #Hmem lapply (mem_map ????? Hmem) * #M1 * #memM1 #eqM H cases (le_to_or_lt_eq … (leb_true_to_le … H)) + [#ltn >(not_eq_to_eqb_false … (lt_to_not_eq … ltn)) normalize + lapply (compare_append … HG) * #G3 * + [* #HG1 #HG2 @(strength_rel … tyN … ltn) HG in ltn; >length_append #ltn @False_ind + @(absurd … ltn) @le_to_not_lt >Hlen // + ] + |#HG21 >(eq_to_eqb_true … HG21) + cut (ty = tyN) + [<(nth_spec ? ty ty ? G2 … Hlen) >HG @nth_spec @HG21] #Hty >Hty + normalize H normalize lapply (compare_append … HG) * #G3 * + [* #H1 @False_ind @(absurd ? Hlen) @sym_not_eq @lt_to_not_eq >H1 + >length_append @(lt_to_le_to_lt n (|G21|)) // @not_le_to_lt + @(leb_false_to_not_le … H) + |cases G3 + [>append_nil * #H1 @False_ind @(absurd ? Hlen)

associative_append @trel // + ] + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tapp … ty1) [@(HindM … eqG HN0) |@(HindN … eqG HN0)] + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tlambda … ty1) @(HindM (ty1::G1) … HN0) >eqG // + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tvec … ty1) + [>length_map @Hlen + |#M #Hmem lapply (mem_map ????? Hmem) * #a * -Hmem #Hmem #eqM HM1 @tlambda @Hind // + |#HM1 >HM1 @tvec // #N #HN lapply(mem_map ????? HN) + * #a * #mema #eqN H3 @tvec + [H2 >length_append >length_append @eq_f // + |#M2 #Hmem cases (mem_append ???? Hmem) -Hmem #Hmem + [@Hv >H2 @mem_append_l1 // + |cases Hmem + [#HM2 >HM2 -HM2 @(Hind N … H1) >H2 @mem_append_l2 %1 // + |-Hmem #Hmem @Hv >H2 @mem_append_l2 %2 // + ] + ] + ] + ] +qed. + diff --git a/matita/matita/lib/finite_lambda/typing_old.ma b/matita/matita/lib/finite_lambda/typing_old.ma new file mode 100644 index 000000000..7ab0bdbfd --- /dev/null +++ b/matita/matita/lib/finite_lambda/typing_old.ma @@ -0,0 +1,236 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "finite_lambda/finite_lambda.ma". + + +(****************************************************************) + +inductive TJ (O: Type[0]) (D:O → FinSet): list (FType O) → T O D → FType O → Prop ≝ + | tval: ∀G,o,a. TJ O D G (Val O D o a) (atom O o) + | trel: ∀G1,ty,G2,n. length ? G1 = n → TJ O D (G1@ty::G2) (Rel O D n) ty + | tapp: ∀G,M,N,ty1,ty2. TJ O D G M (arrow O ty1 ty2) → TJ O D G N ty1 → + TJ O D G (App O D M N) ty2 + | tlambda: ∀G,M,ty1,ty2. TJ O D (ty1::G) M ty2 → + TJ O D G (Lambda O D ty1 M) (arrow O ty1 ty2) + | tvec: ∀G,v,ty1,ty2. + (|v| = |enum (FinSet_of_FType O D ty1)|) → + (∀M. mem ? M v → TJ O D G M ty2) → + TJ O D G (Vec O D ty1 v) (arrow O ty1 ty2). + +lemma wt_to_T: ∀O,D,G,ty,a.TJ O D G (to_T O D ty a) ty. +#O #D #G #ty elim ty + [#o #a normalize @tval + |#ty1 #ty2 #Hind1 #Hind2 normalize * #v #Hv @tvec + [length_map >length_map // + |#M elim v + [normalize @False_ind |#a #v1 #Hind3 * [#eqM >eqM @Hind2 |@Hind3]] + ] + ] +qed. + +lemma inv_rel: ∀O,D,G,n,ty. + TJ O D G (Rel O D n) ty → ∃G1,G2.|G1|=n∧G=G1@ty::G2. +#O #D #G #n #ty #Hrel inversion Hrel + [#G1 #o #a #_ #H destruct + |#G1 #ty1 #G2 #n1 #H1 #H2 #H3 #H4 destruct %{G1} %{G2} /2/ + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tlambda: ∀O,D,G,M,ty1,ty2,ty3. + TJ O D G (Lambda O D ty1 M) (arrow O ty2 ty3) → + ty1 = ty2 ∧ TJ O D (ty2::G) M ty3. +#O #D #G #M #ty1 #ty2 #ty3 #Hlam inversion Hlam + [#G1 #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct % // + |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct + ] +qed. + +lemma inv_tvec: ∀O,D,G,v,ty1,ty2,ty3. + TJ O D G (Vec O D ty1 v) (arrow O ty2 ty3) → + (|v| = |enum (FinSet_of_FType O D ty1)|) ∧ + (∀M. mem ? M v → TJ O D G M ty3). +#O #D #G #v #ty1 #ty2 #ty3 #Hvec inversion Hvec + [#G #o #a #_ #H destruct + |#G1 #ty #G2 #n #_ #_ #H destruct + |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct + |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct + |#G1 #v1 #ty4 #ty5 #Hv #Hmem #_ #_ #H #H1 destruct % // @Hmem + ] +qed. + +(* could be generalized *) +lemma weak_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@G2) (Rel O D n) ty1 → + TJ O D (G1@ty2::G2) (Rel O D (S n)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |* #H3 #H4 >H4 >append_cons length_append >length_append

H3 >length_append normalize + >plus_n_Sm >associative_plus @eq_f // + ] +qed. + +lemma strength_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n → + TJ O D (G1@ty2::G2) (Rel O D n) ty1 → + TJ O D (G1@G2) (Rel O D (n-1)) ty1. +#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel) +* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2) +* #G5 * + [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4 + @(absurd … H4) @le_to_not_lt // + |lapply G5 -G5 * + [>append_nil normalize * #H3 #H4 destruct @False_ind @(absurd … HG1) + @le_to_not_lt // + |#ty3 #G5 * #H3 normalize #H4 destruct (H4) H3 >length_append >length_append normalize Hcase normalize + [lapply (compare_append … H) * #G6 * + [* #H1 #H2 >H2 H1 >length_append >length_append >length_append // + |cases G6 + [* >append_nil normalize #H1 #H2

H1 >length_append // + |#ty1 #G7 * #H @False_ind @(absurd … (leb_true_to_le … Hcase)) + @lt_to_not_le H >length_append normalize // + ] + ] + |lapply (compare_append … H) * #G6 * + [* #H1 @False_ind @(absurd ?? (leb_false_to_not_le … Hcase)) H1 + >length_append normalize // + |* cases G6 + [>append_nil normalize #H1 @False_ind + @(absurd ?? (leb_false_to_not_le … Hcase)) H1 // + |#ty1 #G7 #H1 normalize #H2 destruct >associative_append @trel // + ] + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #G3 + #Heq @(tapp … (HindM … Heq) (HindN … Heq)) + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #G3 #Heq @tlambda @(HindM (ty1::G1)) + >Heq // + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #G3 #H1 @tvec + [>length_map // + |#M #Hmem lapply (mem_map ????? Hmem) * #M1 * #memM1 #eqM H cases (le_to_or_lt_eq … (leb_true_to_le … H)) + [#ltn >(not_eq_to_eqb_false … (lt_to_not_eq … ltn)) normalize + lapply (compare_append … HG) * #G3 * + [* #HG1 #HG2 @(strength_rel … tyN … ltn) HG in ltn; >length_append #ltn @False_ind + @(absurd … ltn) @le_to_not_lt >Hlen // + ] + |#HG21 >(eq_to_eqb_true … HG21) + cut (ty = tyN) + [<(nth_spec ? ty ty ? G2 … Hlen) >HG @nth_spec @HG21] #Hty >Hty + normalize H normalize lapply (compare_append … HG) * #G3 * + [* #H1 @False_ind @(absurd ? Hlen) @sym_not_eq @lt_to_not_eq >H1 + >length_append @(lt_to_le_to_lt n (|G21|)) // @not_le_to_lt + @(leb_false_to_not_le … H) + |cases G3 + [>append_nil * #H1 @False_ind @(absurd ? Hlen)

associative_append @trel // + ] + ] + ] + |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tapp … ty1) [@(HindM … eqG HN0) |@(HindN … eqG HN0)] + |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tlambda … ty1) @(HindM (ty1::G1) … HN0) >eqG // + |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #N0 #tyN0 #eqG + #HN0 normalize @(tvec … ty1) + [>length_map @Hlen + |#M #Hmem lapply (mem_map ????? Hmem) * #a * -Hmem #Hmem #eqM HM1 + @tvec // #N #HN lapply(mem_map ????? HN) * #a * #mema #eqN H3 @tvec + [H2 >length_append >length_append @eq_f // + |#M2 #Hmem cases (mem_append ???? Hmem) -Hmem #Hmem + [@Hv >H2 @mem_append_l1 // + |cases Hmem + [#HM2 >HM2 -HM2 @(Hind N … H1) >H2 @mem_append_l2 %1 // + |-Hmem #Hmem @Hv >H2 @mem_append_l2 %2 // + ] + ] + ] + ] +qed. + -- 2.39.2