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.*)
+
(****************************************************************)
naxiom A: nat → nat → Prop.
naxiom R: nat → nat → nat → Prop.
naxiom conv: T → T → Prop.
-nlemma mah: ∀A,i. lift A i = lift_aux A 0 i.
-//; nqed.
-
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 B a)
+ 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;
- (* napplyS start non va *)
- ncut (S (length T L) = ((length T L)+0+1)); ##[//##] #Heq;
- napplyS start;/2/;
+ #tjN1; nnormalize; ndestruct;
+ napplyS start; /2/;
##]
##|#G; #P; #Q; #R; #i; #tjP; #tjR; #Hind1; #Hind2;
#G1; #D; #N; ncases D; nnormalize;
#G1; #D; #N; #Heq; #tjN; nnormalize;
napply (prod … Ax);
##[/2/;
- ##|(* metas not found *)
- napplyS (Hind2 G1 (P::D) N );
- nnormalize;
+ ##|ncut (S (length T D) = (length T D)+1); ##[//##] #Heq1;
+ nrewrite < Heq1;
+ napply (Hind2 ? (P::D));nnormalize;//;
##]
##|#G; #P; #Q; #R; #S; #tjP; #tjS; #Hind1; #Hind2;
+ #G1; #D; #N; #Heq; #tjN; nnormalize in Hind1 ⊢ %;
+ nrewrite > (plus_n_O (length ? D)) in ⊢ (? ? ? (? ? % ?));
+ nrewrite > (subst_lemma R S N ? 0);
+ napplyS app; /2/;
+ ##|#G; #P; #Q; #R; #i; #tjR; #tjProd; #Hind1; #Hind2;
#G1; #D; #N; #Heq; #tjN; nnormalize;
- ncheck app.
-
+ napplyS abs;
+ ##[nnormalize in Hind2; /2/;
+ ##|(* napplyS (Hind1 G1 (P::D) N ? tjN); sistemare *)
+ ngeneralize in match (Hind1 G1 (P::D) N ? tjN);
+ ##[#H; nnormalize in H; napplyS H;##|nnormalize; //##]
+ ##|##]
+ ##|
+
-
-
-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;//;
- ##]
-
-
-
+