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.*)
42 record pts : Type[0] ≝ {
44 Re: nat → nat → nat → Prop;
48 inductive TJ (p: pts): list T → T → T → Prop ≝
49 | ax : ∀i,j. Ax p i j → TJ p (nil T) (Sort i) (Sort j)
50 | start: ∀G.∀A.∀i.TJ p G A (Sort i) →
51 TJ p (A::G) (Rel 0) (lift A 0 1)
53 TJ p G A B → TJ p G C (Sort i) →
54 TJ p (C::G) (lift A 0 1) (lift B 0 1)
55 | prod: ∀G.∀A,B.∀i,j,k. Re p i j k →
56 TJ p G A (Sort i) → TJ p (A::G) B (Sort j) →
57 TJ p G (Prod A B) (Sort k)
59 TJ p G F (Prod A B) → TJ p G a A →
60 TJ p G (App F a) (subst B 0 a)
62 TJ p (A::G) b B → TJ p G (Prod A B) (Sort i) →
63 TJ p G (Lambda A b) (Prod A B)
64 | conv: ∀G.∀A,B,C.∀i. Co p B C →
65 TJ p G A B → TJ p G C (Sort i) → TJ p G A C
67 TJ p G A B → TJ p G B (Sort i) → TJ p G (D A) B.
69 interpretation "generic type judgement" 'TJT P G A B = (TJ P G A B).
71 notation "hvbox( G break ⊢ _{P} A break : B)"
72 non associative with precedence 45
73 for @{'TJT $P $G $A $B}.
75 (* ninverter TJ_inv2 for TJ (%?%) : Prop. *)
77 (**** definitions ****)
79 inductive Glegal (P:pts) (G: list T) : Prop ≝
80 glegalk : ∀A,B. G ⊢_{P} A : B → Glegal P G.
82 inductive Gterm (P:pts) (G: list T) (A:T) : Prop ≝
83 | is_term: ∀B.G ⊢_{P} A:B → Gterm P G A
84 | is_type: ∀B.G ⊢_{P} B:A → Gterm P G A.
86 inductive Gtype (P:pts) (G: list T) (A:T) : Prop ≝
87 gtypek: ∀i.G ⊢_{P} A : Sort i → Gtype P G A.
89 inductive Gelement (P:pts) (G:list T) (A:T) : Prop ≝
90 gelementk: ∀B.G ⊢_{P} A:B → Gtype P G B → Gelement P G A.
92 inductive Tlegal (P:pts) (A:T) : Prop ≝
93 tlegalk: ∀G. Gterm P G A → Tlegal P A.
96 ndefinition Glegal ≝ λG: list T.∃A,B:T.TJ G A B .
98 ndefinition Gterm ≝ λG: list T.λA.∃B.TJ G A B ∨ TJ G B A.
100 ndefinition Gtype ≝ λG: list T.λA.∃i.TJ G A (Sort i).
102 ndefinition Gelement ≝ λG: list T.λA.∃B.TJ G A B ∨ Gtype G B.
104 ndefinition Tlegal ≝ λA:T.∃G: list T.Gterm G A.
108 ntheorem free_var1: ∀G.∀A,B,C. TJ G A B →
110 #G; #i; #j; #axij; #Gleg; ncases Gleg;
111 #A; #B; #tjAB; nelim tjAB; /2/; (* bello *) nqed.
114 theorem start_lemma1: ∀P,G,i,j.
115 Ax P i j → Glegal P G → G ⊢_{P} Sort i: Sort j.
116 #P #G #i #j #axij #Gleg (cases Gleg)
117 #A #B #tjAB (elim tjAB) /2/
120 theorem start_rel: ∀P,G,A,C,n,i,q.
121 G ⊢_{P} C: Sort q → G ⊢_{P} Rel n: lift A 0 i →
122 C::G ⊢_{P} Rel (S n): lift A 0 (S i).
123 #P #G #A #C #n #i #p #tjC #tjn
124 (applyS (weak P G (Rel n))) //.
127 theorem start_lemma2: ∀P,G.
128 Glegal P G → ∀n. n < |G| → G ⊢_{P} Rel n: lift (nth n T G (Rel O)) 0 (S n).
129 #P #G #Gleg (cases Gleg) #A #B #tjAB (elim tjAB) /2/
130 [#i #j #axij #p normalize #abs @(False_ind) @(absurd … abs) //
131 |#G #A #i #tjA #Hind #m (cases m) /2/
132 #p #Hle @start_rel // @Hind @le_S_S_to_le @Hle
133 |#G #A #B #C #i #tjAB #tjC #Hind1 #_ #m (cases m)
134 /2/ #p #Hle @start_rel // @Hind1 @le_S_S_to_le @Hle
138 axiom conv_subst: ∀T,P,Q,N,i.Co T P Q → Co T P[i := N] Q[i := N].
140 theorem substitution_tj:
141 ∀P,E.∀A,B,M. E ⊢_{P} M:B → ∀G,D.∀N. E = D@A::G → G ⊢_{P} N:A →
142 ((substl D N)@G) ⊢_{P} M[|D| := N]: B[|D| := N].
143 #Pts #E #A #B #M #tjMB (elim tjMB)
144 [normalize #i #j #k #G #D #N (cases D)
145 [normalize #isnil destruct
146 |#P #L normalize #isnil destruct
148 |#G #A1 #i #tjA #Hind #G1 #D (cases D)
149 [#N #Heq #tjN >(delift (lift N O O) A1 O O O ??) //
150 (normalize in Heq) destruct /2/
151 |#H #L #N1 #Heq (normalize in Heq)
152 #tjN1 normalize destruct; (applyS start) /2/
154 |#G #P #Q #R #i #tjP #tjR #Hind1 #Hind2 #G1 #D #N
156 [#Heq destruct #tjN //
157 |#H #L #Heq #tjN1 destruct;
158 (* napplyS weak non va *)
159 (cut (S (length T L) = (length T L)+0+1)) [//]
160 #Hee (applyS weak) /2/
162 |#G #P #Q #i #j #k #Ax #tjP #tjQ #Hind1 #Hind2
163 #G1 #D #N #Heq #tjN normalize @(prod … Ax);
165 |(cut (S (length T D) = (length T D)+1)) [//]
166 #Heq1 <Heq1 @(Hind2 ? (P::D)) normalize //
168 |#G #P #Q #R #S #tjP #tjS #Hind1 #Hind2
169 #G1 #D #N #Heq #tjN (normalize in Hind1 ⊢ %)
170 >(plus_n_O (length ? D)) in ⊢ (? ? ? ? (? ? % ?))
171 >(subst_lemma R S N ? 0) (applyS app) /2/
172 |#G #P #Q #R #i #tjR #tjProd #Hind1 #Hind2
173 #G1 #D #N #Heq #tjN normalize
175 [normalize in Hind2 /2/
176 |(* napplyS (Hind1 G1 (P::D) N ? tjN); sistemare *)
177 generalize in match (Hind1 G1 (P::D) N ? tjN);
178 [#H (normalize in H) (applyS H) | normalize // ]
180 |#G #P #Q #R #i #convQR #tjP #tjQ #Hind1 #Hind2
182 @(conv …(conv_subst … convQR) ? (Hind2 …)) // @Hind1 //
183 |#G #P #Q #i #tjP #tjQ #Hind1 #Hind2
184 #G1 #D #N #Heq #tjN @dummy /2/
188 lemma tj_subst_0: ∀P,G,v,w. G ⊢_{P} v : w → ∀t,u. w :: G ⊢_{P} t : u →
189 G ⊢_{P} t[0≝v] : u[0≝v].
190 #P #G #v #w #Hv #t #u #Ht
191 lapply (substitution_tj … Ht ? ([]) … Hv) normalize //