(* Basic_2A1: uses: lsx_intro_alt *)
lemma rsx_intro_lpxs (h) (G):
- â\88\80L1,T. (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\80L1,T. (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«.
/4 width=1 by lpx_lpxs, rsx_intro/ qed-.
(* Basic_2A1: uses: lsx_lpxs_trans *)
lemma rsx_lpxs_trans (h) (G):
- â\88\80L1,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L1â¦\84 →
- â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â¬\88*[h] L2 â\86\92 G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L2â¦\84.
+ â\88\80L1,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL1â\9d« →
+ â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â¬\88*[h] L2 â\86\92 G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL2â\9d«.
#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
/2 width=3 by rsx_lpx_trans/
qed-.
(* Eliminators with unbound rt-computation for full local environments ******)
lemma rsx_ind_lpxs_reqx (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\80L1. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L1â¦\84 →
- â\88\80L0. â¦\83G,L1â¦\84 ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+ â\88\80L1. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL1â\9d« →
+ â\88\80L0. â\9dªG,L1â\9d« ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
#h #G #T #Q #IH #L1 #H @(rsx_ind … H) -L1
#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
@IH -IH /3 width=3 by rsx_lpxs_trans, rsx_reqx_trans/ -HL1 #K2 #HLK2 #HnLK2
(* Basic_2A1: uses: lsx_ind_alt *)
lemma rsx_ind_lpxs (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 #IH #L #HL
@(rsx_ind_lpxs_reqx … IH … HL) -IH -HL // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
fact rsx_bind_lpxs_aux (h) (G):
- â\88\80p,I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83L1â¦\84 →
- â\88\80Y,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Yâ¦\84 →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ â\88\80p,I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªL1â\9d« →
+ â\88\80Y,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªYâ\9d« →
+ ∀L2. Y = L2.ⓑ[I]V → ❪G,L1❫ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ[p,I]V.T] 𝐒❪L2❫.
#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
(* Basic_2A1: uses: lsx_bind *)
lemma rsx_bind (h) (G):
- â\88\80p,I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Lâ¦\84 →
- â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L.â\93\91{I}Vâ¦\84 →
- G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ â\88\80p,I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªLâ\9d« →
+ â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL.â\93\91[I]Vâ\9d« →
+ G ⊢ ⬈*[h,ⓑ[p,I]V.T] 𝐒❪L❫.
/2 width=3 by rsx_bind_lpxs_aux/ qed.
(* Basic_2A1: uses: lsx_flat_lpxs *)
lemma rsx_flat_lpxs (h) (G):
- â\88\80I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83L1â¦\84 →
- â\88\80L2,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L2â¦\84 â\86\92 â¦\83G,L1â¦\84 ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄.
+ â\88\80I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªL1â\9d« →
+ â\88\80L2,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL2â\9d« â\86\92 â\9dªG,L1â\9d« ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓕ[I]V.T] 𝐒❪L2❫.
#h #G #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
#L1 #HL1 #IHL1 #L2 #T #H @(rsx_ind_lpxs … H) -L2
#L2 #HL2 #IHL2 #HL12 @rsx_intro_lpxs
(* Basic_2A1: uses: lsx_flat *)
lemma rsx_flat (h) (G):
- â\88\80I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Lâ¦\84 →
- â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Lâ¦\84 â\86\92 G â\8a¢ â¬\88*[h,â\93\95{I}V.T] ð\9d\90\92â¦\83Lâ¦\84.
+ â\88\80I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªLâ\9d« →
+ â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªLâ\9d« â\86\92 G â\8a¢ â¬\88*[h,â\93\95[I]V.T] ð\9d\90\92â\9dªLâ\9d«.
/2 width=3 by rsx_flat_lpxs/ qed.
fact rsx_bind_lpxs_void_aux (h) (G):
- â\88\80p,I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83L1â¦\84 →
- â\88\80Y,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83Yâ¦\84 →
- â\88\80L2. Y = L2.â\93§ â\86\92 â¦\83G,L1â¦\84 ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ â\88\80p,I,L1,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªL1â\9d« →
+ â\88\80Y,T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªYâ\9d« →
+ â\88\80L2. Y = L2.â\93§ â\86\92 â\9dªG,L1â\9d« ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ[p,I]V.T] 𝐒❪L2❫.
#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
qed-.
lemma rsx_bind_void (h) (G):
- â\88\80p,I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â¦\83Lâ¦\84 →
- â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â¦\83L.â\93§â¦\84 →
- G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ â\88\80p,I,L,V. G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªLâ\9d« →
+ â\88\80T. G â\8a¢ â¬\88*[h,T] ð\9d\90\92â\9dªL.â\93§â\9d« →
+ G ⊢ ⬈*[h,ⓑ[p,I]V.T] 𝐒❪L❫.
/2 width=3 by rsx_bind_lpxs_void_aux/ qed.