nlet rec substl (G:list T) (N:T) : list T ≝
match G with
[ nil ⇒ nil T
- | cons A D ⇒ ((subst_aux A (length T D) N)::(substl D N))
+ | cons A D ⇒ ((subst A (length T D) N)::(substl D N))
].
(*
nlemma substl_cons: ∀A,N.∀G.
substl (A::G) N = (subst_aux A (length T G) N)::(substl G N).
//; nqed.
+*)
+(*
nlemma length_cons: ∀A.∀G. length T (A::G) = length T G + 1.
-/2/; nqed.
-*)
+/2/; nqed.*)
(****************************************************************)
naxiom R: nat → nat → nat → Prop.
naxiom conv: T → T → Prop.
-nlemma mah: ∀A,i. lift A i = lift_aux A 0 i.
-//; nqed.
-
-ncheck subst.
-
ninductive TJ: list T → T → T → Prop ≝
| ax : ∀i,j. A i j → TJ (nil T) (Sort i) (Sort j)
- | start: ∀G.∀A.∀i.TJ G A (Sort i) → TJ (A::G) (Rel 0) (lift A 1)
+ | start: ∀G.∀A.∀i.TJ G A (Sort i) → TJ (A::G) (Rel 0) (lift A 0 1)
| weak: ∀G.∀A,B,C.∀i.
- TJ G A B → TJ G C (Sort i) → TJ (C::G) (lift A 1) (lift B 1)
+ TJ G A B → TJ G C (Sort i) → TJ (C::G) (lift A 0 1) (lift B 0 1)
| prod: ∀G.∀A,B.∀i,j,k. R i j k →
TJ G A (Sort i) → TJ (A::G) B (Sort j) → TJ G (Prod A B) (Sort k)
| app: ∀G.∀F,A,B,a.
- TJ G F (Prod A B) → TJ G a A → TJ G (App F a) (subst a B)
+ TJ G F (Prod A B) → TJ G a A → TJ G (App F a) (subst B 0 a)
| abs: ∀G.∀A,B,b.∀i.
TJ (A::G) b B → TJ G (Prod A B) (Sort i) → TJ G (Lambda A b) (Prod A B)
| conv: ∀G.∀A,B,C.∀i. conv B C →
(* bello *) nqed.
ntheorem start_rel: ∀G.∀A.∀C.∀n,i,q.
-G ⊢ C: Sort q → G ⊢ Rel n: lift A i → (C::G) ⊢ Rel (S n): lift A (S i).
+G ⊢ C: Sort q → G ⊢ Rel n: lift A 0 i → (C::G) ⊢ Rel (S n): lift A 0 (S i).
#G; #A; #C; #n; #i; #p; #tjC; #tjn;
napplyS (weak G (Rel n));//. (* bello *)
(*
nqed.
ntheorem start_lemma2: ∀G.
-Glegal G → ∀n. n < |G| → G ⊢ Rel n: lift (nth n T G (Rel O)) (S n).
+Glegal G → ∀n. n < |G| → G ⊢ Rel n: lift (nth n T G (Rel O)) 0 (S n).
#G; #Gleg; ncases Gleg; #A; #B; #tjAB; nelim tjAB; /2/;
##[#i; #j; #axij; #p; nnormalize; #abs; napply False_ind;
napply (absurd … abs); //;
##]
##|#G; #A1; #i; #tjA; #Hind; #G1; #D; ncases D;
##[#N; #Heq; #tjN;
- nrewrite > (delift (lift N O) A1 O O O ??); //;
+ nrewrite > (delift (lift N O O) A1 O O O ??); //;
nnormalize in Heq; ndestruct;/2/;
##|#H; #L; #N1; #Heq; nnormalize in Heq;
- #tjN1; nnormalize; ndestruct;
+ #tjN1; nnormalize; ndestruct;
+ ncheck( let clause_829:
+ ∀x1947: ?.
+ ∀x1948: ?.
+ ∀x1949: ?.
+ ∀x1950: ?.
+ ∀x1951: ?.
+ eq T (lift (subst x1947 (plus x1948 x1949) x1950) x1949 x1951)
+ (subst (lift x1947 x1949 x1951) (plus x1948 (plus x1949 x1951))
+ x1950)
+ ≝ λx1947:?.
+ λx1948:?.
+ λx1949:?.
+ λx1950:?.
+ λx1951:?.
+ rewrite_l nat (plus (plus x1948 x1949) x1951)
+ (λx:nat.
+ eq T (lift (subst x1947 (plus x1948 x1949) x1950) x1949 x1951)
+ (subst (lift x1947 x1949 x1951) x x1950))
+ (lift_subst_ijk x1950 x1947 x1951 x1948 x1949)
+ (plus x1948 (plus x1949 x1951))
+ (associative_plus x1948 x1949 x1951) in
+ let clause_60: ∀x156: ?. eq nat (S x156) (plus x156 (S O))
+ ≝ λx156:?.
+ rewrite_r nat (plus x156 O) (λx:nat. eq nat (S x) (plus x156 (S O)))
+ (plus_n_Sm x156 O) x156 (plus_n_O x156) in
+ let clause_996 :
+ eq Type
+ (TJ (cons T (subst ? ? ?) ?) (Rel O)
+ (subst (lift ? O (S O)) (plus ? (S O)) ?))
+ (TJ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1))
+ ≝ rewrite_l nat (S (length T L))
+ (λx:nat.
+ eq Type
+ (TJ
+ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) x N1))
+ (TJ
+ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1)))
+ (refl Type
+ (TJ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1)))
+ (plus (length T L) (S O)) (clause_60 (length T L)) in
+ let clause_995:
+ eq Type
+ (TJ (cons T (subst ? ? ?) ?) (Rel O)
+ (subst (lift ? O (S O)) (plus ? (plus O (S O))) ?))
+ (TJ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1))
+ ≝ rewrite_l nat (S O)
+ (λx:nat.
+ eq Type
+ (TJ (cons T (subst ? ? ?) ?) (Rel O)
+ (subst (lift ? O (S O)) (plus ? x) ?))
+ (TJ
+ (cons T (subst H (length T L) N1)
+ (append T (substl L N1) G1)) (Rel O)
+ (subst (lift H O (S O)) (S (length T L)) N1))) clause_996
+ (plus O (S O)) (plus_O_n (S O)) in
+ rewrite_r T
+ (subst (lift ? O (S O)) (plus ? (plus O (S O))) ?)
+ (λx:T.
+ eq Type
+ (TJ (cons T (subst ? (plus ? O) ?) ?) (Rel O) x)
+ (TJ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1)))
+ (rewrite_l nat ?
+ (λx:nat.
+ eq Type
+ (TJ (cons T (subst ? x ?) ?) (Rel O)
+ (subst (lift ? O (S O)) (plus ? (plus O (S O))) ?))
+ (TJ
+ (cons T (subst H (length T L) N1) (append T (substl L N1) G1))
+ (Rel O) (subst (lift H O (S O)) (S (length T L)) N1)))
+ clause_995 (plus ? O) (plus_n_O ?))
+ (lift (subst ? (plus ? O) ?) O (S O))
+ (clause_829 ? ? O ? (S O))
+).
+ napplyS start;
+ (* napplyS start; *)
(* napplyS start non va *)
ncut (S (length T L) = ((length T L)+0+1)); ##[//##] #Heq;
+ ncheck start.
napplyS start;/2/;
##]
##|#G; #P; #Q; #R; #i; #tjP; #tjR; #Hind1; #Hind2;
#G1; #D; #N; #Heq; #tjN; nnormalize;
napply (prod … Ax);
##[/2/;
- ##|ncheck (Hind2 G1 (P::D) N ? tjN).
- ncut (S (length T D) = (length T D)+1); ##[//##] #Heq1;
- nrewrite < Heq1;
+ ##|ncut (S (length T D) = (length T D)+1); ##[//##] #Heq1;
+ nrewrite < Heq1;
napply (Hind2 ? (P::D));//;
##]
##|#G; #P; #Q; #R; #S; #tjP; #tjS; #Hind1; #Hind2;
#G1; #D; #N; #Heq; #tjN; nnormalize;
+ ncheck (
+ (subst (subst_aux S (length T D) N)
+ (subst_aux R (length T D) N))
+ ).
+ napplyS (app (substl D N@G1) (subst_aux P (length T D) N) A (subst_aux R (length T D) N) (subst_aux S (length T D) N) ?).
nlapply (subst_lemma R S N (length ? D) 0); #sl;
nrewrite < (plus_n_O ?) in sl; #sl;
nrewrite > sl;
+ nauto demod;
napply app; nnormalize in Hind1;/2/;
##|
-
-
-
-ntheorem substitution_tj:
-∀E.∀A,B,M.TJ E M B → ∀G,D.∀N. E = D@A::G → TJ G N A →
-∀k.length ? D = k →
- TJ ((substl D N)@G) (subst_aux M k N) (subst_aux B k N).
-#E; #A; #B; #M; #tjMB; nelim tjMB;
- ##[nnormalize; (* /3/; *)
- ##|#G; #A1; #i; #tjA; #Hind;
- #G1; #D; ncases D;
- ##[#N; #Heq; #tjN; #k; nnormalize in ⊢ (% → ?); #kO;
- nrewrite < kO;
- nrewrite > (delift (lift N O) A1 O O O ??); //;
- nnormalize in Heq; ndestruct;/2/;
- ##|#H; #L; #N1; #Heq; nnormalize in Heq;
- #tjN1; #k; #len; nnormalize in len;
- nrewrite < len;
- nnormalize; ndestruct;
- (* porcherie *)
- ncut (S (length T L) = S ((length T L)+0)); ##[//##] #Heq;
- nrewrite > Heq;
- nrewrite < (lift_subst_aux_k N1 H (length T L) O);
- nrewrite < (plus_n_O (length T L));
- napply (start (substl L N1@G1) (subst_aux H (length T L) N1) i ?).
- napply Hind;//;
- ##]
-
-
-
+
| Prod: T → T → T (* type, body *)
.
-nlet rec lift_aux t k p ≝
+nlet rec lift t k p ≝
match t with
[ Sort n ⇒ Sort n
| Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
- | App m n ⇒ App (lift_aux m k p) (lift_aux n k p)
- | Lambda m n ⇒ Lambda (lift_aux m k p) (lift_aux n (k+1) p)
- | Prod m n ⇒ Prod (lift_aux m k p) (lift_aux n (k+1) p)
+ | App m n ⇒ App (lift m k p) (lift n k p)
+ | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
+ | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
].
-ndefinition lift ≝ λt.λp.lift_aux t 0 p.
+(*
+ndefinition lift ≝ λt.λp.lift_aux t 0 p.*)
-notation "↑ \sup n ( M )" non associative with precedence 70 for @{'Lift $n $M}.
-notation "↑ \sub k \sup n ( M )" non associative with precedence 70 for @{'Lift_aux $n $k $M}.
+notation "↑ \sup n ( M )" non associative with precedence 70 for @{'Lift O $M}.
+notation "↑ \sub k \sup n ( M )" non associative with precedence 70 for @{'Lift $n $k $M}.
-interpretation "Lift" 'Lift n M = (lift M n).
-interpretation "Lift_aux" 'Lift_aux n k M = (lift_aux M k n).
+(* interpretation "Lift" 'Lift n M = (lift M n). *)
+interpretation "Lift" 'Lift n k M = (lift M k n).
-nlet rec subst_aux t k a ≝
+nlet rec subst t k a ≝
match t with
[ Sort n ⇒ Sort n
| Rel n ⇒ if_then_else T (leb (S n) k) (Rel n)
- (if_then_else T (eqb n k) (lift a n) (Rel (n-1)))
- | App m n ⇒ App (subst_aux m k a) (subst_aux n k a)
- | Lambda m n ⇒ Lambda (subst_aux m k a) (subst_aux n (k+1) a)
- | Prod m n ⇒ Prod (subst_aux m k a) (subst_aux n (k+1) a)
+ (if_then_else T (eqb n k) (lift a 0 n) (Rel (n-1)))
+ | App m n ⇒ App (subst m k a) (subst n k a)
+ | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
+ | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
].
+(* meglio non definire
ndefinition subst ≝ λa.λt.subst_aux t 0 a.
-
notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
-notation "M [ k ← N]" non associative with precedence 90 for @{'Subst_aux $M $k $N}.
+*)
+
+notation "M [ k ← N]" non associative with precedence 90 for @{'Subst $M $k $N}.
-interpretation "Subst" 'Subst N M = (subst N M).
-interpretation "Subst_aux" 'Subst_aux M k N = (subst_aux M k N).
+(* interpretation "Subst" 'Subst N M = (subst N M). *)
+interpretation "Subst" 'Subst M k N = (subst M k N).
(*** properties of lift and subst ***)
-nlemma lift_aux_0: ∀t:T.∀k. lift_aux t k 0 = t.
+nlemma lift_0: ∀t:T.∀k. lift t k 0 = t.
#t; nelim t; nnormalize; //; #n; #k; ncases (leb (S n) k);
nnormalize;//;nqed.
-nlemma lift_0: ∀t:T. lift t 0 = t.
-#t; nelim t; nnormalize; //; nqed.
+(* nlemma lift_0: ∀t:T. lift t 0 = t.
+#t; nelim t; nnormalize; //; nqed. *)
-nlemma lift_sort: ∀i,k. lift (Sort i) k = Sort i.
+nlemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
//; nqed.
-nlemma lift_rel: ∀i,k. lift (Rel i) k = Rel (i+k).
+nlemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
//; nqed.
-nlemma lift_rel1: ∀i.lift (Rel i) 1 = Rel (S i).
-#i; nchange with (lift (Rel i) 1 = Rel (1 + i)); //; nqed.
+nlemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
+#i; nchange with (lift (Rel i) 0 1 = Rel (1 + i)); //; nqed.
-nlemma lift_lift_aux: ∀t.∀i,j.j ≤ i → ∀h,k.
-lift_aux (lift_aux t k i) (j+k) h = lift_aux t k (i+h).
+nlemma lift_lift: ∀t.∀i,j.j ≤ i → ∀h,k.
+ lift (lift t k i) (j+k) h = lift t k (i+h).
#t; #i; #j; #h; nelim t; nnormalize; //; #n; #h;#k;
napply (leb_elim (S n) k); #Hnk;nnormalize;
##[nrewrite > (le_to_leb_true (S n) (j+k) ?);nnormalize;/2/;
##]
nqed.
-nlemma lift_lift_aux1: ∀t.∀i,j,k. lift_aux (lift_aux t k j) k i = lift_aux t k (j+i).
+nlemma lift_lift1: ∀t.∀i,j,k.
+ lift(lift t k j) k i = lift t k (j+i).
#t;/3/; nqed.
-nlemma lift_lift_aux2: ∀t.∀i,j,k. lift_aux (lift_aux t k j) (j+k) i = lift_aux t k (j+i).
+nlemma lift_lift2: ∀t.∀i,j,k.
+ lift (lift t k j) (j+k) i = lift t k (j+i).
#t; /2/; nqed.
+(*
nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
-nnormalize; //; nqed.
+nnormalize; //; nqed. *)
-nlemma subst_lift_aux_k: ∀A,B.∀k.
- subst_aux (lift_aux B k 1) k A = B.
+nlemma subst_lift_k: ∀A,B.∀k. subst (lift B k 1) k A = B.
#A; #B; nelim B; nnormalize; /2/; #n; #k;
napply (leb_elim (S n) k); nnormalize; #Hnk;
##[nrewrite > (le_to_leb_true ?? Hnk);nnormalize;//;
##]
nqed.
+(*
nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
-nnormalize; //; nqed.
-
-nlemma subst_aux_sort: ∀A.∀n,k. subst_aux (Sort n) k A = Sort n.
-//; nqed.
+nnormalize; //; nqed. *)
-nlemma subst_sort: ∀A.∀n. subst A (Sort n) = Sort n.
+nlemma subst_sort: ∀A.∀n,k. subst (Sort n) k A = Sort n.
//; nqed.
-nlemma subst_rel: ∀A.subst A (Rel O) = A.
+nlemma subst_rel: ∀A.subst (Rel 0) 0 A = A.
nnormalize; //; nqed.
nlemma subst_rel1: ∀A.∀k,i. i < k →
- subst_aux (Rel i) k A = Rel i.
+ subst (Rel i) k A = Rel i.
#A; #k; #i; nnormalize; #ltik;
nrewrite > (le_to_leb_true (S i) k ?); //; nqed.
-nlemma subst_rel2: ∀A.∀k. subst_aux (Rel k) k A = lift A k.
+nlemma subst_rel2: ∀A.∀k.
+ subst (Rel k) k A = lift A 0 k.
#A; #k; nnormalize;
nrewrite > (lt_to_leb_false (S k) k ?); //;
nrewrite > (eq_to_eqb_true … (refl …)); //;
nqed.
nlemma subst_rel3: ∀A.∀k,i. k < i →
- subst_aux (Rel i) k A = Rel (i-1).
+ subst (Rel i) k A = Rel (i-1).
#A; #k; #i; nnormalize; #ltik;
nrewrite > (lt_to_leb_false (S i) k ?); /2/;
nrewrite > (not_eq_to_eqb_false i k ?); //;
napply nmk; #eqik; nelim (lt_to_not_eq … (ltik …)); /2/;
nqed.
-nlemma lift_subst_aux_ijk: ∀A,B.∀i,j,k.
- lift_aux (subst_aux B (j+k) A) k i = subst_aux (lift_aux B k i) (j+k+i) A.
+nlemma lift_subst_ijk: ∀A,B.∀i,j,k.
+ lift (subst B (j+k) A) k i = subst (lift B k i) (j+k+i) A.
#A; #B; #i; #j; nelim B; nnormalize; /2/; #n; #k;
napply (leb_elim (S n) (j + k)); nnormalize; #Hnjk;
##[nelim (leb (S n) k);
##[nrewrite > (lt_to_leb_false (S n) k ?);
##[ncut (j+k+i = n+i);##[//;##] #Heq;
nrewrite > Heq; nrewrite > (subst_rel2 A ?);
- nnormalize; napplyS lift_lift_aux;//;
+ nnormalize; napplyS lift_lift;//;
##|/2/;
##]
##|ncut (j + k < n);
nqed.
ntheorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
- subst_aux (lift_aux B i (S k)) j A = (lift_aux B i k).
+ subst (lift B i (S k)) j A = (lift B i k).
#A; #B; nelim B; nnormalize; /2/;
##[##2,3,4: #T; #T0; #Hind1; #Hind2; #i; #j; #k; #leij; #lejk;
napply eq_f2;/2/; napply Hind2;
(********************* substitution lemma ***********************)
nlemma subst_lemma: ∀A,B,C.∀k,i.
- subst_aux (subst_aux A i B) (k+i) C =
- subst_aux (subst_aux A (S (k+i)) C) i (subst_aux B k C).
+ subst (subst A i B) (k+i) C =
+ subst (subst A (S (k+i)) C) i (subst B k C).
#A; #B; #C; #k; nelim A; nnormalize;//; (* WOW *)
#n; #i; napply (leb_elim (S n) i); #Hle;
##[ncut (n < k+i); ##[/2/##] #ltn; (* lento *)
nrewrite > (le_to_leb_true i (k+i) ?); //;
nrewrite > (subst_rel2 …); nnormalize;
napply symmetric_eq;
- napplyS (lift_subst_aux_ijk C B i k O);
+ napplyS (lift_subst_ijk C B i k O);
##|napply (leb_elim (S (n-1)) (k+i)); #nk;
##[nrewrite > (subst_rel1 C (k+i) (n-1) nk);
nrewrite > (le_to_leb_true n (k+i) ?);
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