SN … (lpx h G) (reqx T).
interpretation
- "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
- 'PRedTySNStrong h T G L = (rsx h G T L).
+ "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
+ 'PRedTySNStrong h T G L = (rsx h G T L).
(* Basic eliminators ********************************************************)
(* Basic_2A1: uses: lsx_ind *)
-lemma rsx_ind (h) (G) (T) (Q:predicate lenv):
- (∀L1. G ⊢ ⬈*[h,T] 𝐒❪L1❫ →
- (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
- Q L1
+lemma rsx_ind (h) (G) (T) (Q:predicate …):
+ (∀L1. G ⊢ ⬈*𝐒[h,T] L1 →
+ (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
) →
- ∀L. G ⊢ ⬈*[h,T] 𝐒❪L❫ → Q L.
+ ∀L. G ⊢ ⬈*𝐒[h,T] L → 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.
- (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒❪L2❫) →
- G ⊢ ⬈*[h,T] 𝐒❪L1❫.
+ (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*𝐒[h,T] L2) →
+ G ⊢ ⬈*𝐒[h,T] L1.
/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 ⊢ ⬈*[h,V] 𝐒❪L❫.
+ ∀I,L,V,T. G ⊢ ⬈*𝐒[h,②[I]V.T] L →
+ G ⊢ ⬈*𝐒[h,V] L.
#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 ⊢ ⬈*[h,T] 𝐒❪L❫.
+ ∀I,L,V,T. G ⊢ ⬈*𝐒[h,ⓕ[I]V.T] L →
+ G ⊢ ⬈*𝐒[h,T] L.
#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):
- ∀L. G ⊢ ⬈*[h,#0] 𝐒❪L❫ →
- ∀I,K,V. L = K.ⓑ[I]V → G ⊢ ⬈*[h,V] 𝐒❪K❫.
+ ∀L. G ⊢ ⬈*𝐒[h,#0] L →
+ ∀I,K,V. L = K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,V] 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):
- ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒❪K.ⓑ[I]V❫ → G ⊢ ⬈*[h,V] 𝐒❪K❫.
+ ∀I,K,V. G ⊢ ⬈*𝐒[h,#0] K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,V] K.
/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❫ →
- ∧∧ G ⊢ ⬈*[h,V] 𝐒❪L❫ & G ⊢ ⬈*[h,T] 𝐒❪L❫.
+ ∀I,L,V,T. G ⊢ ⬈*𝐒[h,ⓕ[I]V.T] L →
+ ∧∧ G ⊢ ⬈*𝐒[h,V] L & G ⊢ ⬈*𝐒[h,T] L.
/3 width=3 by rsx_fwd_pair_sn, rsx_fwd_flat_dx, conj/ qed-.
(* Basic_2A1: removed theorems 9: