--- /dev/null
+(*
+ ||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 "lambda/subst.ma".
+include "basics/list.ma".
+
+
+(*************************** substl *****************************)
+
+let rec substl (G:list T) (N:T) : list T ≝
+ match G with
+ [ nil ⇒ nil T
+ | 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.*)
+
+(****************************************************************)
+
+axiom A: nat → nat → Prop.
+axiom R: nat → nat → nat → Prop.
+axiom conv: T → T → Prop.
+
+inductive 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 0 1)
+ | weak: ∀G.∀A,B,C.∀i.
+ 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 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 →
+ TJ G A B → TJ G B (Sort i) → TJ G A C
+ | dummy: ∀G.∀A,B.∀i.
+ TJ G A B → TJ G B (Sort i) → TJ G (D A) B.
+
+notation "hvbox(G break ⊢ A : B)" non associative with precedence 50 for @{'TJ $G $A $B}.
+interpretation "type judgement" 'TJ G A B = (TJ G A B).
+
+(* ninverter TJ_inv2 for TJ (%?%) : Prop. *)
+
+(**** definitions ****)
+
+inductive Glegal (G: list T) : Prop ≝
+glegalk : ∀A,B. G ⊢ A : B → Glegal G.
+
+inductive Gterm (G: list T) (A:T) : Prop ≝
+ | is_term: ∀B.G ⊢ A:B → Gterm G A
+ | is_type: ∀B.G ⊢ B:A → Gterm G A.
+
+inductive Gtype (G: list T) (A:T) : Prop ≝
+gtypek: ∀i.G ⊢ A : Sort i → Gtype G A.
+
+inductive Gelement (G:list T) (A:T) : Prop ≝
+gelementk: ∀B.G ⊢ A:B → Gtype G B → Gelement G A.
+
+inductive Tlegal (A:T) : Prop ≝
+tlegalk: ∀G. Gterm G A → Tlegal A.
+
+(*
+ndefinition Glegal ≝ λG: list T.∃A,B:T.TJ G A B .
+
+ndefinition Gterm ≝ λG: list T.λA.∃B.TJ G A B ∨ TJ G B A.
+
+ndefinition Gtype ≝ λG: list T.λA.∃i.TJ G A (Sort i).
+
+ndefinition Gelement ≝ λG: list T.λA.∃B.TJ G A B ∨ Gtype G B.
+
+ndefinition Tlegal ≝ λA:T.∃G: list T.Gterm G A.
+*)
+
+(*
+ntheorem free_var1: ∀G.∀A,B,C. TJ G A B →
+subst C A
+#G; #i; #j; #axij; #Gleg; ncases Gleg;
+#A; #B; #tjAB; nelim tjAB; /2/; (* bello *) nqed.
+*)
+
+theorem start_lemma1: ∀G.∀i,j.
+A i j → Glegal G → G ⊢ Sort i: Sort j.
+#G #i #j #axij #Gleg (cases Gleg)
+#A #B #tjAB (elim tjAB) /2/
+(* bello *) qed.
+
+theorem start_rel: ∀G.∀A.∀C.∀n,i,q.
+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
+ (applyS (weak G (Rel n))) //. (* bello *)
+ (*
+ nrewrite > (plus_n_O i);
+ nrewrite > (plus_n_Sm i O);
+ nrewrite < (lift_lift A 1 i);
+ nrewrite > (plus_n_O n); nrewrite > (plus_n_Sm n O);
+ applyS (weak G (Rel n) (lift A i) C p tjn tjC). *)
+qed.
+
+theorem start_lemma2: ∀G.
+Glegal G → ∀n. n < |G| → G ⊢ Rel n: lift (nth n T G (Rel O)) 0 (S n).
+#G #Gleg (cases Gleg) #A #B #tjAB (elim tjAB) /2/
+ [#i #j #axij #p normalize #abs @False_ind @(absurd … abs) //
+ |#G #A #i #tjA #Hind #m (cases m) /2/
+ #p #Hle @start_rel // @Hind @le_S_S_to_le @Hle
+ |#G #A #B #C #i #tjAB #tjC #Hind1 #_ #m (cases m)
+ /2/ #p #Hle @start_rel // @Hind1 @le_S_S_to_le @Hle
+ ]
+qed.
+
+(*
+nlet rec TJm G D on D : Prop ≝
+ match D with
+ [ nil ⇒ True
+ | cons A D1 ⇒ TJ G (Rel 0) A ∧ TJm G D1
+ ].
+
+nlemma tjm1: ∀G,D.∀A. TJm G (A::D) → TJ G (Rel 0) A.
+#G; #D; #A; *; //; nqed.
+
+ntheorem transitivity_tj: ∀D.∀A,B. TJ D A B →
+ ∀G. Glegal G → TJm G D → TJ G A B.
+#D; #A; #B; #tjAB; nelim tjAB;
+ ##[/2/;
+ ##|/2/;
+ ##|#E; #T; #T0; #T1; #n; #tjT; #tjT1; #H; #H1; #G; #HlegG;
+ #tjGcons;
+ napply weak;
+*)
+(*
+ntheorem substitution_tj:
+∀G.∀A,B,N,M.TJ (A::G) M B → TJ G N A →
+ TJ G (subst N M) (subst N B).
+#G;#A;#B;#N;#M;#tjM;
+ napply (TJ_inv2 (A::G) M B);
+ ##[nnormalize; /3/;
+ ##|#G; #A; #N; #tjA; #Hind; #Heq;
+ ndestruct;//;
+ ##|#G; #A; #B; #C; #n; #tjA; #tjC; #Hind1; #Hind2; #Heq;
+ ndestruct;//;
+ ##|nnormalize; #E; #A; #B; #i; #j; #k;
+ #Ax; #tjA; #tjB; #Hind1; #_;
+ #Heq; #HeqB; #tjN; napply (prod ?????? Ax);
+ ##[/2/;
+ ##|nnormalize; napplyS weak;
+
+*)
+
+
+axiom conv_subst: ∀P,Q,N,i.conv P Q → conv P[i := N] Q[i := N].
+
+theorem substitution_tj:
+∀E.∀A,B,M. E ⊢M:B → ∀G,D.∀N. E = D@A::G → G ⊢ N:A →
+ ((substl D N)@G) ⊢ M[|D| := N]: B[|D| := N].
+#E #A #B #M #tjMB (elim tjMB)
+ [normalize #i #j #k #G #D #N (cases D)
+ [normalize #isnil destruct
+ |#P #L normalize #isnil destruct
+ ]
+ |#G #A1 #i #tjA #Hind #G1 #D (cases D)
+ [#N #Heq #tjN >(delift (lift N O O) A1 O O O ??) //
+ (normalize in Heq) destruct /2/
+ |#H #L #N1 #Heq (normalize in Heq)
+ #tjN1 normalize destruct; (applyS start) /2/
+ ]
+ |#G #P #Q #R #i #tjP #tjR #Hind1 #Hind2 #G1 #D #N
+ (cases D) normalize
+ [#Heq destruct #tjN //
+ |#H #L #Heq #tjN1 destruct;
+ (* napplyS weak non va *)
+ (cut (S (length T L) = (length T L)+0+1)) [//]
+ #Hee (applyS weak) /2/
+ ]
+ |#G #P #Q #i #j #k #Ax #tjP #tjQ #Hind1 #Hind2
+ #G1 #D #N #Heq #tjN normalize @(prod … Ax);
+ [/2/
+ |(cut (S (length T D) = (length T D)+1)) [//]
+ #Heq1 <Heq1 @(Hind2 ? (P::D)) normalize //
+ ]
+ |#G #P #Q #R #S #tjP #tjS #Hind1 #Hind2
+ #G1 #D #N #Heq #tjN (normalize in Hind1 ⊢ %)
+ >(plus_n_O (length ? D)) in ⊢ (? ? ? (? ? % ?))
+ >(subst_lemma R S N ? 0) (applyS app) /2/
+ |#G #P #Q #R #i #tjR #tjProd #Hind1 #Hind2
+ #G1 #D #N #Heq #tjN normalize
+ (applyS abs)
+ [normalize in Hind2 /2/
+ |(* napplyS (Hind1 G1 (P::D) N ? tjN); sistemare *)
+ generalize in match (Hind1 G1 (P::D) N ? tjN);
+ [#H (normalize in H) (applyS H) | normalize // ]
+ ]
+ |#G #P #Q #R #i #convQR #tjP #tjQ #Hind1 #Hind2
+ #G1 #D #N #Heq #tjN
+ @(conv …(conv_subst … convQR) ? (Hind2 …)) // @Hind1 //
+ |#G #P #Q #i #tjP #tjQ #Hind1 #Hind2
+ #G1 #D #N #Heq #tjN @dummy /2/
+ ]
+qed.
projects / helm.git / blobdiff