(* Forward lemmas involving same top term structure *************************)
-lemma cpxs_fwd_cnx: â\88\80h,o,G,L,T. â¦\83G, Lâ¦\84 â\8a¢ â\9e¡[h, o] ð\9d\90\8dâ¦\83Tâ¦\84 â\86\92 â\88\80U. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] U → T ≂ U.
+lemma cpxs_fwd_cnx: â\88\80h,o,G,L,T. â¦\83G, Lâ¦\84 â\8a¢ â¬\88[h, o] ð\9d\90\8dâ¦\83Tâ¦\84 â\86\92 â\88\80U. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] U → T ≂ U.
#h #o #G #L #T #HT #U #H
>(cpxs_inv_cnx1 … H HT) -G -L -T //
qed-.
-lemma cpxs_fwd_sort: â\88\80h,o,G,L,U,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡*[h, o] U →
- â\8b\86s â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\8b\86(next h s) â\9e¡*[h, o] U.
+lemma cpxs_fwd_sort: â\88\80h,o,G,L,U,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â¬\88*[h, o] U →
+ â\8b\86s â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\8b\86(next h s) â¬\88*[h, o] U.
#h #o #G #L #U #s #H
elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus … n) -n
[ #s #_ #H -d destruct /2 width=1 by or_introl/
qed-.
(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: â\88\80h,o,a,G,L,V,W,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V.â\93\9b{a}W.T â\9e¡*[h, o] U →
- â\93\90V.â\93\9b{a}W.T â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}â\93\9dW.V.T â\9e¡*[h, o] U.
+lemma cpxs_fwd_beta: â\88\80h,o,a,G,L,V,W,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V.â\93\9b{a}W.T â¬\88*[h, o] U →
+ â\93\90V.â\93\9b{a}W.T â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}â\93\9dW.V.T â¬\88*[h, o] U.
#h #o #a #G #L #V #W #T #U #H
elim (cpxs_inv_appl1 … H) -H *
[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
(* Note: probably this is an inversion lemma *)
lemma cpxs_fwd_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
∀V2. ⬆[0, i + 1] V1 ≡ V2 →
- â\88\80U. â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡*[h, o] U →
- #i â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ V2 â\9e¡*[h, o] U.
+ â\88\80U. â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88*[h, o] U →
+ #i â\89\82 U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ V2 â¬\88*[h, o] U.
#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
/4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
qed-.
-lemma cpxs_fwd_theta: â\88\80h,o,a,G,L,V1,V,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}V.T â\9e¡*[h, o] U →
+lemma cpxs_fwd_theta: â\88\80h,o,a,G,L,V1,V,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}V.T â¬\88*[h, o] U →
∀V2. ⬆[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
- â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}V.â\93\90V2.T â\9e¡*[h, o] U.
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}V.â\93\90V2.T â¬\88*[h, o] U.
#h #o #a #G #L #V1 #V #T #U #H #V2 #HV12
elim (cpxs_inv_appl1 … H) -H *
[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
]
qed-.
-lemma cpxs_fwd_cast: â\88\80h,o,G,L,W,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dW.T â\9e¡*[h, o] U →
- â\88¨â\88¨ â\93\9dW. T â\89\82 U | â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] U | â¦\83G, Lâ¦\84 â\8a¢ W â\9e¡*[h, o] U.
+lemma cpxs_fwd_cast: â\88\80h,o,G,L,W,T,U. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dW.T â¬\88*[h, o] U →
+ â\88¨â\88¨ â\93\9dW. T â\89\82 U | â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] U | â¦\83G, Lâ¦\84 â\8a¢ W â¬\88*[h, o] U.
#h #o #G #L #W #T #U #H
elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
#W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/