(* Basic_2A1: uses: lsx_intro_alt *)
lemma rdsx_intro_lpxs (h) (G):
- ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) →
- G ⊢ ⬈*[h, T] 𝐒⦃L1⦄.
+ ∀L1,T. (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
/4 width=1 by lpx_lpxs, rdsx_intro/ qed-.
(* Basic_2A1: uses: lsx_lpxs_trans *)
lemma rdsx_lpxs_trans (h) (G):
- ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄.
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
/2 width=3 by rdsx_lpx_trans/
qed-.
lemma rdsx_ind_lpxs_rdeq (h) (G):
∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- ∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+ ∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G,L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
#h #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
@IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
(* Basic_2A1: uses: lsx_ind_alt *)
lemma rdsx_ind_lpxs (h) (G):
∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
Q L1
) →
- ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → Q L.
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
#h #G #T #Q #IH #L #HL
@(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
fact rdsx_bind_lpxs_aux (h) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
(* Basic_2A1: uses: lsx_bind *)
lemma rdsx_bind (h) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓑ{I}V⦄ →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓑ{I}V⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_bind_lpxs_aux/ qed.
(* Basic_2A1: uses: lsx_flat_lpxs *)
lemma rdsx_flat_lpxs (h) (G):
- ∀I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀L2,T. G ⊢ ⬈*[h, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L2⦄.
+ ∀I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2,T. G ⊢ ⬈*[h,T] 𝐒⦃L2⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄.
#h #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2
#L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs
(* Basic_2A1: uses: lsx_flat *)
lemma rdsx_flat (h) (G):
- ∀I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄.
+ ∀I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_flat_lpxs/ qed.
fact rdsx_bind_lpxs_void_aux (h) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓧ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct
qed-.
lemma rdsx_bind_void (h) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄ →
- G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
/2 width=3 by rdsx_bind_lpxs_void_aux/ qed.