(* Basic_2A1: uses: lsx_ind *)
lemma rsx_ind (h) (G) (T) (Q:predicate lenv):
- (â\88\80L1. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L1â¦\84 →
- (â\88\80L2. â¦\83G,L1â¦\84 ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (â\88\80L1. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL1â\9d« →
+ (â\88\80L2. â\9dªG,L1â\9d« ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- â\88\80L. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Lâ¦\84 → Q L.
+ â\88\80L. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªLâ\9d« → Q L.
#h #G #T #Q #H0 #L1 #H elim H -L1
/5 width=1 by SN_intro/
qed-.
(* Basic_2A1: uses: lsx_intro *)
lemma rsx_intro (h) (G) (T):
∀L1.
- (â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â¬\88[h] L2 â\86\92 (L1 â\89\9b[T] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L2â¦\84) →
- G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L1â¦\84.
+ (â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â¬\88[h] L2 â\86\92 (L1 â\89\9b[T] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL2â\9d«) →
+ G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL1â\9d«.
/5 width=1 by SN_intro/ qed.
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
lemma rsx_fwd_pair_sn (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h,②{I}V.T] 𝐒⦃L⦄ →
- G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Lâ¦\84.
+ ∀I,L,V,T. G ⊢ ⬈*[h,②[I]V.T] 𝐒❪L❫ →
+ G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªLâ\9d«.
#h #G #I #L #V #T #H
@(rsx_ind … H) -L #L1 #_ #IHL1
@rsx_intro #L2 #HL12 #HnL12
(* Basic_2A1: uses: lsx_fwd_flat_dx *)
lemma rsx_fwd_flat_dx (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
- G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Lâ¦\84.
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ[I]V.T] 𝐒❪L❫ →
+ G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªLâ\9d«.
#h #G #I #L #V #T #H
@(rsx_ind … H) -L #L1 #_ #IHL1
@rsx_intro #L2 #HL12 #HnL12
qed-.
fact rsx_fwd_pair_aux (h) (G):
- â\88\80L. G â\8a¢ â¬\88*[h,#0] ð\9d\90\92â¦\83Lâ¦\84 →
- ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+ â\88\80L. G â\8a¢ â¬\88*[h,#0] ð\9d\90\92â\9dªLâ\9d« →
+ ∀I,K,V. L = K.ⓑ[I]V → G ⊢ ⬈*[h,V] 𝐒❪K❫.
#h #G #L #H
@(rsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
/5 width=5 by lpx_pair, rsx_intro, reqx_fwd_zero_pair/
qed-.
lemma rsx_fwd_pair (h) (G):
- â\88\80I,K,V. G â\8a¢ â¬\88*[h,#0] ð\9d\90\92â¦\83K.â\93\91{I}Vâ¦\84 â\86\92 G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Kâ¦\84.
+ â\88\80I,K,V. G â\8a¢ â¬\88*[h,#0] ð\9d\90\92â\9dªK.â\93\91[I]Vâ\9d« â\86\92 G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªKâ\9d«.
/2 width=4 by rsx_fwd_pair_aux/ qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: lsx_inv_flat *)
lemma rsx_inv_flat (h) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
- â\88§â\88§ G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Lâ¦\84 & G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Lâ¦\84.
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ[I]V.T] 𝐒❪L❫ →
+ â\88§â\88§ G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªLâ\9d« & G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªLâ\9d«.
/3 width=3 by rsx_fwd_pair_sn, rsx_fwd_flat_dx, conj/ qed-.
(* Basic_2A1: removed theorems 9: