--- /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 "finite_lambda/reduction.ma".
+
+
+(****************************************************************)
+
+inductive TJ (O: Type[0]) (D:O → FinSet): list (FType O) → T O D → FType O → Prop ≝
+ | tval: ∀G,o,a. TJ O D G (Val O D o a) (atom O o)
+ | trel: ∀G1,ty,G2,n. length ? G1 = n → TJ O D (G1@ty::G2) (Rel O D n) ty
+ | tapp: ∀G,M,N,ty1,ty2. TJ O D G M (arrow O ty1 ty2) → TJ O D G N ty1 →
+ TJ O D G (App O D M N) ty2
+ | tlambda: ∀G,M,ty1,ty2. TJ O D (ty1::G) M ty2 →
+ TJ O D G (Lambda O D ty1 M) (arrow O ty1 ty2)
+ | tvec: ∀G,v,ty1,ty2.
+ (|v| = |enum (FinSet_of_FType O D ty1)|) →
+ (∀M. mem ? M v → TJ O D G M ty2) →
+ TJ O D G (Vec O D ty1 v) (arrow O ty1 ty2).
+
+lemma wt_to_T: ∀O,D,G,ty,a.TJ O D G (to_T O D ty a) ty.
+#O #D #G #ty elim ty
+ [#o #a normalize @tval
+ |#ty1 #ty2 #Hind1 #Hind2 normalize * #v #Hv @tvec
+ [<Hv >length_map >length_map //
+ |#M elim v
+ [normalize @False_ind |#a #v1 #Hind3 * [#eqM >eqM @Hind2 |@Hind3]]
+ ]
+ ]
+qed.
+
+lemma inv_rel: ∀O,D,G,n,ty.
+ TJ O D G (Rel O D n) ty → ∃G1,G2.|G1|=n∧G=G1@ty::G2.
+#O #D #G #n #ty #Hrel inversion Hrel
+ [#G1 #o #a #_ #H destruct
+ |#G1 #ty1 #G2 #n1 #H1 #H2 #H3 #H4 destruct %{G1} %{G2} /2/
+ |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct
+ |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct
+ |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct
+ ]
+qed.
+
+lemma inv_tlambda: ∀O,D,G,M,ty1,ty2,ty3.
+ TJ O D G (Lambda O D ty1 M) (arrow O ty2 ty3) →
+ ty1 = ty2 ∧ TJ O D (ty2::G) M ty3.
+#O #D #G #M #ty1 #ty2 #ty3 #Hlam inversion Hlam
+ [#G1 #o #a #_ #H destruct
+ |#G1 #ty #G2 #n #_ #_ #H destruct
+ |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct
+ |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct % //
+ |#G1 #v #ty3 #ty4 #_ #_ #_ #_ #H destruct
+ ]
+qed.
+
+lemma inv_tvec: ∀O,D,G,v,ty1,ty2,ty3.
+ TJ O D G (Vec O D ty1 v) (arrow O ty2 ty3) →
+ (|v| = |enum (FinSet_of_FType O D ty1)|) ∧
+ (∀M. mem ? M v → TJ O D G M ty3).
+#O #D #G #v #ty1 #ty2 #ty3 #Hvec inversion Hvec
+ [#G #o #a #_ #H destruct
+ |#G1 #ty #G2 #n #_ #_ #H destruct
+ |#G1 #M0 #N #ty1 #ty2 #_ #_ #_ #_ #_ #H destruct
+ |#G1 #M0 #ty4 #ty5 #HM0 #_ #_ #H #H1 destruct
+ |#G1 #v1 #ty4 #ty5 #Hv #Hmem #_ #_ #H #H1 destruct % // @Hmem
+ ]
+qed.
+
+(* could be generalized *)
+lemma weak_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n →
+ TJ O D (G1@G2) (Rel O D n) ty1 →
+ TJ O D (G1@ty2::G2) (Rel O D (S n)) ty1.
+#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel)
+* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2)
+* #G5 *
+ [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4
+ @(absurd … H4) @le_to_not_lt //
+ |* #H3 #H4 >H4 >append_cons <associative_append @trel
+ >length_append >length_append <H1 >H3 >length_append normalize
+ >plus_n_Sm >associative_plus @eq_f //
+ ]
+qed.
+
+lemma strength_rel: ∀O,D,G1,G2,ty1,ty2,n. length ? G1 < n →
+ TJ O D (G1@ty2::G2) (Rel O D n) ty1 →
+ TJ O D (G1@G2) (Rel O D (n-1)) ty1.
+#O #D #G1 #G2 #ty1 #ty2 #n #HG1 #Hrel lapply (inv_rel … Hrel)
+* #G3 * #G4 * #H1 #H2 lapply (compare_append … H2)
+* #G5 *
+ [* #H3 @False_ind >H3 in HG1; >length_append >H1 #H4
+ @(absurd … H4) @le_to_not_lt //
+ |lapply G5 -G5 *
+ [>append_nil normalize * #H3 #H4 destruct @False_ind @(absurd … HG1)
+ @le_to_not_lt //
+ |#ty3 #G5 * #H3 normalize #H4 destruct (H4) <associative_append @trel
+ <H1 >H3 >length_append >length_append normalize <plus_minus_associative //
+ ]
+ ]
+qed.
+
+lemma no_matter: ∀O,D,G,N,tyN.
+ TJ O D G N tyN → ∀G1,G2,G3.G=G1@G2 → is_closed O D (|G1|) N →
+ TJ O D (G1@G3) N tyN.
+#O #D #G #N #tyN #HN elim HN -HN -tyN -N -G
+ [#G #o #a #G1 #G2 #G3 #_ #_ @tval
+ |#G #ty #G2 #n #HG #G3 #G4 #G5 #H #HNC normalize
+ lapply (is_closed_rel … HNC) #Hlt lapply (compare_append … H) * #G6 *
+ [* #H1 @False_ind @(absurd ? Hlt) @le_to_not_lt <HG >H1 >length_append //
+ |* cases G6
+ [>append_nil normalize #H1 @False_ind
+ @(absurd ? Hlt) @le_to_not_lt <HG >H1 //
+ |#ty1 #G7 #H1 normalize #H2 destruct >associative_append @trel //
+ ]
+ ]
+ |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #G3
+ #Heq #Hc lapply (is_closed_app … Hc) -Hc * #HMc #HNc
+ @(tapp … (HindM … Heq HMc) (HindN … Heq HNc))
+ |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #G3 #Heq #Hc
+ lapply (is_closed_lam … Hc) -Hc #HMc
+ @tlambda @(HindM (ty1::G1) G2) [>Heq // |@HMc]
+ |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #G3 #H1 #Hc @tvec
+ [>length_map //
+ |#M #Hmem @Hind // lapply (is_closed_vec … Hc) #Hvc @Hvc //
+ ]
+ ]
+qed.
+
+lemma nth_spec: ∀A,a,d,l1,l2,n. |l1| = n → nth n A (l1@a::l2) d = a.
+#A #a #d #l1 elim l1 normalize
+ [#l2 #n #Hn <Hn //
+ |#b #tl #Hind #l2 #m #Hm <Hm normalize @Hind //
+ ]
+qed.
+
+lemma wt_subst_gen: ∀O,D,G,M,tyM.
+ TJ O D G M tyM →
+ ∀G1,G2,N,tyN.G=(G1@tyN::G2) →
+ TJ O D G2 N tyN → is_closed O D 0 N →
+ TJ O D (G1@G2) (subst O D M (|G1|) N) tyM.
+#O #D #G #M #tyM #HM elim HM -HM -tyM -M -G
+ [#G #o #a #G1 #G2 #N #tyN #_ #HG #_ normalize @tval
+ |#G #ty #G2 #n #Hlen #G21 #G22 #N #tyN #HG #HN #HNc
+ normalize cases (true_or_false (leb (|G21|) n))
+ [#H >H cases (le_to_or_lt_eq … (leb_true_to_le … H))
+ [#ltn >(not_eq_to_eqb_false … (lt_to_not_eq … ltn)) normalize
+ lapply (compare_append … HG) * #G3 *
+ [* #HG1 #HG2 @(strength_rel … tyN … ltn) <HG @trel @Hlen
+ |* #HG >HG in ltn; >length_append #ltn @False_ind
+ @(absurd … ltn) @le_to_not_lt >Hlen //
+ ]
+ |#HG21 >(eq_to_eqb_true … HG21)
+ cut (ty = tyN)
+ [<(nth_spec ? ty ty ? G2 … Hlen) >HG @nth_spec @HG21] #Hty >Hty
+ normalize <HG21 @(no_matter ????? HN []) //
+ ]
+ |#H >H normalize lapply (compare_append … HG) * #G3 *
+ [* #H1 @False_ind @(absurd ? Hlen) @sym_not_eq @lt_to_not_eq >H1
+ >length_append @(lt_to_le_to_lt n (|G21|)) // @not_le_to_lt
+ @(leb_false_to_not_le … H)
+ |cases G3
+ [>append_nil * #H1 @False_ind @(absurd ? Hlen) <H1 @sym_not_eq
+ @lt_to_not_eq @not_le_to_lt @(leb_false_to_not_le … H)
+ |#ty2 #G4 * #H1 normalize #H2 destruct >associative_append @trel //
+ ]
+ ]
+ ]
+ |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #G1 #G2 #N0 #tyN0 #eqG
+ #HN0 #Hc normalize @(tapp … ty1)
+ [@(HindM … eqG HN0 Hc) |@(HindN … eqG HN0 Hc)]
+ |#G #M #ty1 #ty2 #HM #HindM #G1 #G2 #N0 #tyN0 #eqG
+ #HN0 #Hc normalize @(tlambda … ty1) @(HindM (ty1::G1) … HN0) // >eqG //
+ |#G #v #ty1 #ty2 #Hlen #Hv #Hind #G1 #G2 #N0 #tyN0 #eqG
+ #HN0 #Hc normalize @(tvec … ty1)
+ [>length_map @Hlen
+ |#M #Hmem lapply (mem_map ????? Hmem) * #a * -Hmem #Hmem #eqM <eqM
+ @(Hind … Hmem … eqG HN0 Hc)
+ ]
+ ]
+qed.
+
+lemma wt_subst: ∀O,D,M,N,G,ty1,ty2.
+ TJ O D (ty1::G) M ty2 →
+ TJ O D G N ty1 → is_closed O D 0 N →
+ TJ O D G (subst O D M 0 N) ty2.
+#O #D #M #N #G #ty1 #ty2 #HM #HN #Hc @(wt_subst_gen …(ty1::G) … [ ] … HN) //
+qed.
+
+lemma subject_reduction: ∀O,D,M,M1,G,ty.
+ TJ O D G M ty → red O D M M1 → TJ O D G M1 ty.
+#O #D #M #M1 #G #ty #HM lapply M1 -M1 elim HM -HM -ty -G -M
+ [#G #o #a #M1 #Hval elim (red_val ????? Hval)
+ |#G #ty #G1 #n #_ #M1 #Hrel elim (red_rel ???? Hrel)
+ |#G #M #N #ty1 #ty2 #HM #HN #HindM #HindN #M1 #Hred inversion Hred
+ [#P #M0 #N0 #Hc #H1 destruct (H1) #HM1 @(wt_subst … HN) //
+ @(proj2 … (inv_tlambda … HM))
+ |#ty #v #a #M0 #Ha #H1 #H2 destruct @(proj2 … (inv_tvec … HM))
+ @(assoc_to_mem … Ha)
+ |#M2 #M3 #N0 #Hredl #_ #H1 destruct (H1) #eqM1 @(tapp … HN) @HindM @Hredl
+ |#M2 #M3 #N0 #Hredr #_ #H1 destruct (H1) #eqM1 @(tapp … HM) @HindN @Hredr
+ |#ty #N0 #N1 #_ #_ #H1 destruct (H1)
+ |#ty #M0 #H1 destruct (H1)
+ |#ty #N0 #N1 #v #v1 #_ #_ #H1 destruct (H1)
+ ]
+ |#G #P #ty1 #ty2 #HP #Hind #M1 #Hred lapply(red_lambda ????? Hred) *
+ [* #P1 * #HredP #HM1 >HM1 @tlambda @Hind //
+ |#HM1 >HM1 @tvec // #N #HN lapply(mem_map ????? HN)
+ * #a * #mema #eqN <eqN -eqN @(wt_subst …HP) // @wt_to_T
+ ]
+ |#G #v #ty1 #ty2 #Hlen #Hv #Hind #M1 #Hred lapply(red_vec ????? Hred)
+ * #N * #N1 * #v1 * #v2 * * #H1 #H2 #H3 >H3 @tvec
+ [<Hlen >H2 >length_append >length_append @eq_f //
+ |#M2 #Hmem cases (mem_append ???? Hmem) -Hmem #Hmem
+ [@Hv >H2 @mem_append_l1 //
+ |cases Hmem
+ [#HM2 >HM2 -HM2 @(Hind N … H1) >H2 @mem_append_l2 %1 //
+ |-Hmem #Hmem @Hv >H2 @mem_append_l2 %2 //
+ ]
+ ]
+ ]
+ ]
+qed.
+
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