2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda/subst.ma".
13 include "basics/list.ma".
16 (*************************** substl *****************************)
18 let rec substl (G:list T) (N:T) : list T ≝
21 | cons A D ⇒ ((subst A (length T D) N)::(substl D N))
25 nlemma substl_cons: ∀A,N.∀G.
26 substl (A::G) N = (subst_aux A (length T G) N)::(substl G N).
30 (*start: ∀G.∀A.∀i.TJ G A (Sort i) → TJ (A::G) (Rel 0) (lift A 0 1)
32 nlemma length_cons: ∀A.∀G. length T (A::G) = length T G + 1.
35 (****************************************************************)
38 axiom A: nat → nat → Prop.
39 axiom R: nat → nat → nat → Prop.
40 axiom conv: T → T → Prop.*)
44 (R: nat → nat → nat → Prop)
45 (c: T → T → Prop) : list T → T → T → Prop ≝
46 | ax : ∀i,j. S i j → TJ S R c (nil T) (Sort i) (Sort j)
47 | start: ∀G.∀A.∀i.TJ S R c G A (Sort i) →
48 TJ S R c (A::G) (Rel 0) (lift A 0 1)
50 TJ S R c G A B → TJ S R c G C (Sort i) →
51 TJ S R c (C::G) (lift A 0 1) (lift B 0 1)
52 | prod: ∀G.∀A,B.∀i,j,k. R i j k →
53 TJ S R c G A (Sort i) → TJ S R c (A::G) B (Sort j) →
54 TJ S R c G (Prod A B) (Sort k)
56 TJ S R c G F (Prod A B) → TJ S R c G a A →
57 TJ S R c G (App F a) (subst B 0 a)
59 TJ S R c (A::G) b B → TJ S R c G (Prod A B) (Sort i) →
60 TJ S R c G (Lambda A b) (Prod A B)
61 | conv: ∀G.∀A,B,C.∀i. c B C →
62 TJ S R c G A B → TJ S R c G C (Sort i) → TJ S R c G A C
64 TJ S R c G A B → TJ S R c G B (Sort i) → TJ S R c G (D A) B.
66 interpretation "type judgement" 'TJ G A B = (TJ ? ? ? G A B).
68 record pts : Type[0] ≝ {
70 r1: nat → nat → nat → Prop;
75 definition TJ ≝ λp:pts.c p.
77 (* ninverter TJ_inv2 for TJ (%?%) : Prop. *)
79 (**** definitions ****)
81 inductive Glegal (G: list T) : Prop ≝
82 glegalk : ∀A,B. G ⊢ A : B → Glegal G.
84 inductive Gterm (G: list T) (A:T) : Prop ≝
85 | is_term: ∀B.G ⊢ A:B → Gterm G A
86 | is_type: ∀B.G ⊢ B:A → Gterm G A.
88 inductive Gtype (G: list T) (A:T) : Prop ≝
89 gtypek: ∀i.G ⊢ A : Sort i → Gtype G A.
91 inductive Gelement (G:list T) (A:T) : Prop ≝
92 gelementk: ∀B.G ⊢ A:B → Gtype G B → Gelement G A.
94 inductive Tlegal (A:T) : Prop ≝
95 tlegalk: ∀G. Gterm G A → Tlegal A.
98 ndefinition Glegal ≝ λG: list T.∃A,B:T.TJ G A B .
100 ndefinition Gterm ≝ λG: list T.λA.∃B.TJ G A B ∨ TJ G B A.
102 ndefinition Gtype ≝ λG: list T.λA.∃i.TJ G A (Sort i).
104 ndefinition Gelement ≝ λG: list T.λA.∃B.TJ G A B ∨ Gtype G B.
106 ndefinition Tlegal ≝ λA:T.∃G: list T.Gterm G A.
110 ntheorem free_var1: ∀G.∀A,B,C. TJ G A B →
112 #G; #i; #j; #axij; #Gleg; ncases Gleg;
113 #A; #B; #tjAB; nelim tjAB; /2/; (* bello *) nqed.
116 theorem start_lemma1: ∀G.∀i,j.
117 A i j → Glegal G → G ⊢ Sort i: Sort j.
118 #G #i #j #axij #Gleg (cases Gleg)
119 #A #B #tjAB (elim tjAB) /2/
122 theorem start_rel: ∀G.∀A.∀C.∀n,i,q.
123 G ⊢ C: Sort q → G ⊢ Rel n: lift A 0 i → (C::G) ⊢ Rel (S n): lift A 0 (S i).
124 #G #A #C #n #i #p #tjC #tjn
125 (applyS (weak G (Rel n))) //. (* bello *)
127 nrewrite > (plus_n_O i);
128 nrewrite > (plus_n_Sm i O);
129 nrewrite < (lift_lift A 1 i);
130 nrewrite > (plus_n_O n); nrewrite > (plus_n_Sm n O);
131 applyS (weak G (Rel n) (lift A i) C p tjn tjC). *)
134 theorem start_lemma2: ∀G.
135 Glegal G → ∀n. n < |G| → G ⊢ Rel n: lift (nth n T G (Rel O)) 0 (S n).
136 #G #Gleg (cases Gleg) #A #B #tjAB (elim tjAB) /2/
137 [#i #j #axij #p normalize #abs @(False_ind) @(absurd … abs) //
138 |#G #A #i #tjA #Hind #m (cases m) /2/
139 #p #Hle @start_rel // @Hind @le_S_S_to_le @Hle
140 |#G #A #B #C #i #tjAB #tjC #Hind1 #_ #m (cases m)
141 /2/ #p #Hle @start_rel // @Hind1 @le_S_S_to_le @Hle
146 nlet rec TJm G D on D : Prop ≝
149 | cons A D1 ⇒ TJ G (Rel 0) A ∧ TJm G D1
152 nlemma tjm1: ∀G,D.∀A. TJm G (A::D) → TJ G (Rel 0) A.
153 #G; #D; #A; *; //; nqed.
155 ntheorem transitivity_tj: ∀D.∀A,B. TJ D A B →
156 ∀G. Glegal G → TJm G D → TJ G A B.
157 #D; #A; #B; #tjAB; nelim tjAB;
160 ##|#E; #T; #T0; #T1; #n; #tjT; #tjT1; #H; #H1; #G; #HlegG;
165 ntheorem substitution_tj:
166 ∀G.∀A,B,N,M.TJ (A::G) M B → TJ G N A →
167 TJ G (subst N M) (subst N B).
169 napply (TJ_inv2 (A::G) M B);
171 ##|#G; #A; #N; #tjA; #Hind; #Heq;
173 ##|#G; #A; #B; #C; #n; #tjA; #tjC; #Hind1; #Hind2; #Heq;
175 ##|nnormalize; #E; #A; #B; #i; #j; #k;
176 #Ax; #tjA; #tjB; #Hind1; #_;
177 #Heq; #HeqB; #tjN; napply (prod ?????? Ax);
179 ##|nnormalize; napplyS weak;
184 axiom conv_subst: ∀P,Q,N,i.conv P Q → conv P[i := N] Q[i := N].
186 theorem substitution_tj:
187 ∀E.∀A,B,M. E ⊢M:B → ∀G,D.∀N. E = D@A::G → G ⊢ N:A →
188 ((substl D N)@G) ⊢ M[|D| := N]: B[|D| := N].
189 #E #A #B #M #tjMB (elim tjMB)
190 [normalize #i #j #k #G #D #N (cases D)
191 [normalize #isnil destruct
192 |#P #L normalize #isnil destruct
194 |#G #A1 #i #tjA #Hind #G1 #D (cases D)
195 [#N #Heq #tjN >(delift (lift N O O) A1 O O O ??) //
196 (normalize in Heq) destruct /2/
197 |#H #L #N1 #Heq (normalize in Heq)
198 #tjN1 normalize destruct; (applyS start) /2/
200 |#G #P #Q #R #i #tjP #tjR #Hind1 #Hind2 #G1 #D #N
202 [#Heq destruct #tjN //
203 |#H #L #Heq #tjN1 destruct;
204 (* napplyS weak non va *)
205 (cut (S (length T L) = (length T L)+0+1)) [//]
206 #Hee (applyS weak) /2/
208 |#G #P #Q #i #j #k #Ax #tjP #tjQ #Hind1 #Hind2
209 #G1 #D #N #Heq #tjN normalize @(prod … Ax);
211 |(cut (S (length T D) = (length T D)+1)) [//]
212 #Heq1 <Heq1 @(Hind2 ? (P::D)) normalize //
214 |#G #P #Q #R #S #tjP #tjS #Hind1 #Hind2
215 #G1 #D #N #Heq #tjN (normalize in Hind1 ⊢ %)
216 >(plus_n_O (length ? D)) in ⊢ (? ? ? (? ? % ?))
217 >(subst_lemma R S N ? 0) (applyS app) /2/
218 |#G #P #Q #R #i #tjR #tjProd #Hind1 #Hind2
219 #G1 #D #N #Heq #tjN normalize
221 [normalize in Hind2 /2/
222 |(* napplyS (Hind1 G1 (P::D) N ? tjN); sistemare *)
223 generalize in match (Hind1 G1 (P::D) N ? tjN);
224 [#H (normalize in H) (applyS H) | normalize // ]
226 |#G #P #Q #R #i #convQR #tjP #tjQ #Hind1 #Hind2
228 @(conv …(conv_subst … convQR) ? (Hind2 …)) // @Hind1 //
229 |#G #P #Q #i #tjP #tjQ #Hind1 #Hind2
230 #G1 #D #N #Heq #tjN @dummy /2/
234 lemma tj_subst_0: ∀G,v,w. G ⊢ v : w → ∀t,u. w :: G ⊢ t : u →
236 #G #v #w #Hv #t #u #Ht
237 lapply (substitution_tj … Ht ? ([]) … Hv) normalize //