(* Forward lemmas involving same top term constructor ***********************)
-lemma cpxs_fwd_cnx: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ∀U. ⦃h, L⦄ ⊢ T ➡*[g] U → T ≃ U.
+lemma cpxs_fwd_cnx: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≃ U.
#h #g #L #T #HT #U #H
>(cpxs_inv_cnx1 … H HT) -L -T //
qed-.
-lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U →
- ⋆k ≃ U ∨ ⦃h, L⦄ ⊢ ⋆(next h k) ➡*[g] U.
+lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
+ ⋆k ≃ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
#h #g #L #U #k #H
elim (cpxs_inv_sort1 … H) -H #n #l generalize in match k; -k @(nat_ind_plus … n) -n
[ #k #_ #H -l destruct /2 width=1/
qed-.
(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,g,a,L,V,W,T,U. ⦃h, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[g] U →
- ⓐV.ⓛ{a}W.T ≃ U ∨ ⦃h, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[g] U.
+lemma cpxs_fwd_beta: ∀h,g,a,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, g] U →
+ ⓐV.ⓛ{a}W.T ≃ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, g] U.
#h #g #a #L #V #W #T #U #H
elim (cpxs_inv_appl1 … H) -H *
[ #V0 #T0 #_ #_ #H destruct /2 width=1/
(* Note: probably this is an inversion lemma *)
lemma cpxs_fwd_delta: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
∀V2. ⇧[0, i + 1] V1 ≡ V2 →
- ∀U. ⦃h, L⦄ ⊢ #i ➡*[g] U →
- #i ≃ U ∨ ⦃h, L⦄ ⊢ V2 ➡*[g] U.
+ ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, g] U →
+ #i ≃ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
#h #g #I #L #K #V1 #i #HLK #V2 #HV12 #U #H
elim (cpxs_inv_lref1 … H) -H /2 width=1/
* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/
qed-.
-lemma cpxs_fwd_theta: ∀h,g,a,L,V1,V,T,U. ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[g] U →
+lemma cpxs_fwd_theta: ∀h,g,a,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≃ U ∨
- ⦃h, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[g] U.
+ ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U.
#h #g #a #L #V1 #V #T #U #H #V2 #HV12
elim (cpxs_inv_appl1 … H) -H *
[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/
]
qed-.
-lemma cpxs_fwd_cast: ∀h,g,L,W,T,U. ⦃h, L⦄ ⊢ ⓝW.T ➡*[g] U →
- ∨∨ ⓝW. T ≃ U | ⦃h, L⦄ ⊢ T ➡*[g] U | ⦃h, L⦄ ⊢ W ➡*[g] U.
+lemma cpxs_fwd_cast: ∀h,g,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, g] U →
+ ∨∨ ⓝW. T ≃ U | ⦃G, L⦄ ⊢ T ➡*[h, g] U | ⦃G, L⦄ ⊢ W ➡*[h, g] U.
#h #g #L #W #T #U #H
elim (cpxs_inv_cast1 … H) -H /2 width=1/ *
#W0 #T0 #_ #_ #H destruct /2 width=1/