(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_5_3.ma".
include "basic_2/rt_equivalence/cpcs_cpcs.ma".
include "basic_2/dynamic/cnv_preserve_cpcs.ma".
include "basic_2/dynamic/nta.ma".
(* Properties based on preservation *****************************************)
-lemma cnv_cpms_nta (a) (h) (G) (L):
- â\88\80T. â¦\83G,Lâ¦\84 â\8a¢ T ![a,h] â\86\92 â\88\80U.â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[1,h] U â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U.
+lemma cnv_cpms_nta (h) (a) (G) (L):
+ â\88\80T. â\9dªG,Lâ\9d« â\8a¢ T ![h,a] â\86\92 â\88\80U.â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[1,h] U â\86\92 â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U.
/3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
-lemma cnv_nta_sn (a) (h) (G) (L):
- â\88\80T. â¦\83G,Lâ¦\84 â\8a¢ T ![a,h] â\86\92 â\88\83U. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U.
-#a #h #G #L #T #HT
+lemma cnv_nta_sn (h) (a) (G) (L):
+ â\88\80T. â\9dªG,Lâ\9d« â\8a¢ T ![h,a] â\86\92 â\88\83U. â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U.
+#h #a #G #L #T #HT
elim (cnv_fwd_cpm_SO … HT) #U #HTU
/4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
qed-.
(* Basic_1: was: ty3_typecheck *)
-lemma nta_typecheck (a) (h) (G) (L):
- â\88\80T,U. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U â\86\92 â\88\83T0. â¦\83G,Lâ¦\84 â\8a¢ â\93\9dU.T :[a,h] T0.
+lemma nta_typecheck (h) (a) (G) (L):
+ â\88\80T,U. â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U â\86\92 â\88\83T0. â\9dªG,Lâ\9d« â\8a¢ â\93\9dU.T :[h,a] T0.
/3 width=1 by cnv_cast, cnv_nta_sn/ qed-.
(* Basic_1: was: ty3_correct *)
(* Basic_2A1: was: ntaa_fwd_correct *)
-lemma nta_fwd_correct (a) (h) (G) (L):
- â\88\80T,U. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U â\86\92 â\88\83T0. â¦\83G,Lâ¦\84 â\8a¢ U :[a,h] T0.
+lemma nta_fwd_correct (h) (a) (G) (L):
+ â\88\80T,U. â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U â\86\92 â\88\83T0. â\9dªG,Lâ\9d« â\8a¢ U :[h,a] T0.
/3 width=2 by nta_fwd_cnv_dx, cnv_nta_sn/ qed-.
lemma nta_pure_cnv (h) (G) (L):
- â\88\80T,U. â¦\83G,Lâ¦\84 â\8a¢ T :*[h] U →
- â\88\80V. â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.U !*[h] â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.T :*[h] ⓐV.U.
+ â\88\80T,U. â\9dªG,Lâ\9d« â\8a¢ T :[h,ð\9d\9b\9a] U →
+ â\88\80V. â\9dªG,Lâ\9d« â\8a¢ â\93\90V.U ![h,ð\9d\9b\9a] â\86\92 â\9dªG,Lâ\9d« â\8a¢ â\93\90V.T :[h,ð\9d\9b\9a] ⓐV.U.
#h #G #L #T #U #H1 #V #H2
elim (cnv_inv_cast … H1) -H1 #X0 #HU #HT #HUX0 #HTX0
elim (cnv_inv_appl … H2) #n #p #X1 #X2 #_ #HV #_ #HVX1 #HUX2
qed.
(* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
-lemma nta_cpr_conf_lpr (a) (h) (G):
- â\88\80L1,T1,U. â¦\83G,L1â¦\84 â\8a¢ T1 :[a,h] U â\86\92 â\88\80T2. â¦\83G,L1â¦\84 ⊢ T1 ➡[h] T2 →
- â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â\9e¡[h] L2 â\86\92 â¦\83G,L2â¦\84 â\8a¢ T2 :[a,h] U.
-#a #h #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
+lemma nta_cpr_conf_lpr (h) (a) (G):
+ â\88\80L1,T1,U. â\9dªG,L1â\9d« â\8a¢ T1 :[h,a] U â\86\92 â\88\80T2. â\9dªG,L1â\9d« ⊢ T1 ➡[h] T2 →
+ â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â\9e¡[h] L2 â\86\92 â\9dªG,L2â\9d« â\8a¢ T2 :[h,a] U.
+#h #a #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
/3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
qed-.
(* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
-lemma nta_cpr_conf (a) (h) (G) (L):
- â\88\80T1,U. â¦\83G,Lâ¦\84 â\8a¢ T1 :[a,h] U →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[h] T2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cpr_conf (h) (a) (G) (L):
+ â\88\80T1,U. â\9dªG,Lâ\9d« â\8a¢ T1 :[h,a] U →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpm_trans, cpm_cast/
qed-.
(* Note: this is the preservation property *)
(* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
-lemma nta_cprs_conf (a) (h) (G) (L):
- â\88\80T1,U. â¦\83G,Lâ¦\84 â\8a¢ T1 :[a,h] U →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[h] T2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cprs_conf (h) (a) (G) (L):
+ â\88\80T1,U. â\9dªG,Lâ\9d« â\8a¢ T1 :[h,a] U →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpms_trans, cpms_cast/
qed-.
(* Basic_1: uses: ty3_cred_pr2 *)
-lemma nta_lpr_conf (a) (h) (G):
- â\88\80L1,T,U. â¦\83G,L1â¦\84 â\8a¢ T :[a,h] U →
- â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â\9e¡[h] L2 â\86\92 â¦\83G,L2â¦\84 â\8a¢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lpr_conf (h) (a) (G):
+ â\88\80L1,T,U. â\9dªG,L1â\9d« â\8a¢ T :[h,a] U →
+ â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â\9e¡[h] L2 â\86\92 â\9dªG,L2â\9d« â\8a¢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lpr_trans/
qed-.
(* Basic_1: uses: ty3_cred_pr3 *)
-lemma nta_lprs_conf (a) (h) (G):
- â\88\80L1,T,U. â¦\83G,L1â¦\84 â\8a¢ T :[a,h] U →
- â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â\9e¡*[h] L2 â\86\92 â¦\83G,L2â¦\84 â\8a¢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lprs_conf (h) (a) (G):
+ â\88\80L1,T,U. â\9dªG,L1â\9d« â\8a¢ T :[h,a] U →
+ â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â\9e¡*[h] L2 â\86\92 â\9dªG,L2â\9d« â\8a¢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lprs_trans/
qed-.
(* Inversion lemmas based on preservation ***********************************)
-lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
- â\88\80X2. â¦\83G,K.â\93\93Vâ¦\84 â\8a¢ #0 :[a,h] X2 →
- â\88\83â\88\83W,U. â¦\83G,Kâ¦\84 â\8a¢ V :[a,h] W & â¬\86*[1] W â\89\98 U & â¦\83G,K.â\93\93Vâ¦\84 â\8a¢ U â¬\8c*[h] X2 & â¦\83G,K.â\93\93Vâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
+ â\88\80X2. â\9dªG,K.â\93\93Vâ\9d« â\8a¢ #0 :[h,a] X2 →
+ â\88\83â\88\83W,U. â\9dªG,Kâ\9d« â\8a¢ V :[h,a] W & â\87§*[1] W â\89\98 U & â\9dªG,K.â\93\93Vâ\9d« â\8a¢ U â¬\8c*[h] X2 & â\9dªG,K.â\93\93Vâ\9d« â\8a¢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
elim (cpms_inv_delta_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn (a) (h) (G) (L):
- â\88\80X2,i. â¦\83G,Lâ¦\84 â\8a¢ #â\86\91i :[a,h] X2 →
- â\88\83â\88\83I,K,T2,U2. â¦\83G,Kâ¦\84 â\8a¢ #i :[a,h] T2 & â¬\86*[1] T2 â\89\98 U2 & â¦\83G,K.â\93\98{I}â¦\84 â\8a¢ U2 â¬\8c*[h] X2 & â¦\83G,K.â\93\98{I}â¦\84 â\8a¢ X2 ![a,h] & L = K.â\93\98{I}.
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn (h) (a) (G) (L):
+ â\88\80X2,i. â\9dªG,Lâ\9d« â\8a¢ #â\86\91i :[h,a] X2 →
+ â\88\83â\88\83I,K,T2,U2. â\9dªG,Kâ\9d« â\8a¢ #i :[h,a] T2 & â\87§*[1] T2 â\89\98 U2 & â\9dªG,K.â\93\98[I]â\9d« â\8a¢ U2 â¬\8c*[h] X2 & â\9dªG,K.â\93\98[I]â\9d« â\8a¢ X2 ![h,a] & L = K.â\93\98[I].
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
elim (cpms_inv_lref_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L):
- â\88\80X2,i. â¦\83G,Lâ¦\84 â\8a¢ #i :[a,h] X2 →
- â\88¨â\88¨ â\88\83â\88\83K,V,W,U. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G,Kâ¦\84 â\8a¢ V :[a,h] W & â¬\86*[â\86\91i] W â\89\98 U & â¦\83G,Lâ¦\84 â\8a¢ U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![a,h]
- | â\88\83â\88\83K,W,U. â¬\87*[i] L â\89\98 K. â\93\9bW & â¦\83G,Kâ¦\84 â\8a¢ W ![a,h] & â¬\86*[â\86\91i] W â\89\98 U & â¦\83G,Lâ¦\84 â\8a¢ U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
+ â\88\80X2,i. â\9dªG,Lâ\9d« â\8a¢ #i :[h,a] X2 →
+ â\88¨â\88¨ â\88\83â\88\83K,V,W,U. â\87©*[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« â\8a¢ V :[h,a] W & â\87§*[â\86\91i] W â\89\98 U & â\9dªG,Lâ\9d« â\8a¢ U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,a]
+ | â\88\83â\88\83K,W,U. â\87©*[i] L â\89\98 K. â\93\9bW & â\9dªG,Kâ\9d« â\8a¢ W ![h,a] & â\87§*[â\86\91i] W â\89\98 U & â\9dªG,Lâ\9d« â\8a¢ U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,a].
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
elim (cpms_inv_lref1_drops … H2) -H2 *
/4 width=8 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro, or_introl/
| #n #Y #X #U #H #HVU #HUX1 #H0 destruct
lapply (drops_mono … H … HLK) -H #H destruct
- elim (lifts_total V (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #W #HVW
+ elim (lifts_total V (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #W #HVW
lapply (cpms_lifts_bi … HVU (Ⓣ) … L … HVW … HUX1) -U
[ /2 width=2 by drops_isuni_fwd_drop2/ ] #HWX1
/4 width=9 by cprs_div, ex5_3_intro, or_intror/
]
qed-.
-lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
- â\88\80V,T. â¦\83G,Kâ¦\84 â\8a¢ â\93\91{p,I}V.T :[a,h] X2 →
- â\88\83â\88\83U. â¦\83G,Kâ¦\84 â\8a¢ V ![a,h] & â¦\83G,K.â\93\91{I}Vâ¦\84 â\8a¢ T :[a,h] U & â¦\83G,Kâ¦\84 â\8a¢ â\93\91{p,I}V.U â¬\8c*[h] X2 & â¦\83G,Kâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #p * #G #K #X2 #V #T #H
+lemma nta_inv_bind_sn_cnv (h) (a) (p) (I) (G) (K) (X2):
+ â\88\80V,T. â\9dªG,Kâ\9d« â\8a¢ â\93\91[p,I]V.T :[h,a] X2 →
+ â\88\83â\88\83U. â\9dªG,Kâ\9d« â\8a¢ V ![h,a] & â\9dªG,K.â\93\91[I]Vâ\9d« â\8a¢ T :[h,a] U & â\9dªG,Kâ\9d« â\8a¢ â\93\91[p,I]V.U â¬\8c*[h] X2 & â\9dªG,Kâ\9d« â\8a¢ X2 ![h,a].
+#h #a #p * #G #K #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_bind … H1) -H1 #HV #HT
[ elim (cpms_inv_abbr_sn_dx … H2) -H2 *
(* Basic_1: uses: ty3_gen_appl *)
lemma nta_inv_appl_sn (h) (G) (L) (X2):
- â\88\80V,T. â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.T :[h] X2 →
- â\88\83â\88\83p,W,U. â¦\83G,Lâ¦\84 â\8a¢ V :[h] W & â¦\83G,Lâ¦\84 â\8a¢ T :[h] â\93\9b{p}W.U & â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.â\93\9b{p}W.U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![h].
+ â\88\80V,T. â\9dªG,Lâ\9d« â\8a¢ â\93\90V.T :[h,ð\9d\9f\90] X2 →
+ â\88\83â\88\83p,W,U. â\9dªG,Lâ\9d« â\8a¢ V :[h,ð\9d\9f\90] W & â\9dªG,Lâ\9d« â\8a¢ T :[h,ð\9d\9f\90] â\93\9b[p]W.U & â\9dªG,Lâ\9d« â\8a¢ â\93\90V.â\93\9b[p]W.U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,ð\9d\9f\90].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
-elim (cnv_inv_appl_pred … H1) #p #W #U #HV #HT #HVW #HTU
+elim (cnv_inv_appl … H1) #n #p #W #U #H <H -n #HV #HT #HVW #HTU
/5 width=11 by cnv_cpms_nta, cnv_cpms_conf_eq, cpcs_cprs_div, cpms_appl_dx, ex4_3_intro/
qed-.
(* Basic_2A1: uses: nta_fwd_pure1 *)
lemma nta_inv_pure_sn_cnv (h) (G) (L) (X2):
- â\88\80V,T. â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.T :*[h] X2 →
- â\88¨â\88¨ â\88\83â\88\83p,W,U. â¦\83G,Lâ¦\84 â\8a¢ V :*[h] W & â¦\83G,Lâ¦\84 â\8a¢ T :*[h] â\93\9b{p}W.U & â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.â\93\9b{p}W.U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 !*[h]
- | â\88\83â\88\83U. â¦\83G,Lâ¦\84 â\8a¢ T :*[h] U & â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.U !*[h] & â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 !*[h].
+ â\88\80V,T. â\9dªG,Lâ\9d« â\8a¢ â\93\90V.T :[h,ð\9d\9b\9a] X2 →
+ â\88¨â\88¨ â\88\83â\88\83p,W,U. â\9dªG,Lâ\9d« â\8a¢ V :[h,ð\9d\9b\9a] W & â\9dªG,Lâ\9d« â\8a¢ T :[h,ð\9d\9b\9a] â\93\9b[p]W.U & â\9dªG,Lâ\9d« â\8a¢ â\93\90V.â\93\9b[p]W.U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,ð\9d\9b\9a]
+ | â\88\83â\88\83U. â\9dªG,Lâ\9d« â\8a¢ T :[h,ð\9d\9b\9a] U & â\9dªG,Lâ\9d« â\8a¢ â\93\90V.U ![h,ð\9d\9b\9a] & â\9dªG,Lâ\9d« â\8a¢ â\93\90V.U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,ð\9d\9b\9a].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H
elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
[ lapply (cnv_cpms_trans … HT … HT0) #H0
elim (cnv_inv_bind … H0) -H0 #_ #HU
elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
- lapply (cpms_step_dx … HT0 1 (ⓛ{p}W0.U0) ?) -HT0 [ /2 width=1 by cpm_bind/ ] #HT0
+ lapply (cpms_step_dx … HT0 1 (ⓛ[p]W0.U0) ?) -HT0 [ /2 width=1 by cpm_bind/ ] #HT0
lapply (cpms_appl_dx … V V … HT0) [ // ] #HTU0
lapply (cnv_cpms_conf_eq … H1 … HTU0 … H) -H1 -H -HTU0 #HU0X1
/4 width=8 by cnv_cpms_nta, cpcs_cprs_div, ex4_3_intro, or_introl/
qed-.
(* Basic_2A1: uses: nta_inv_cast1 *)
-lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
- â\88\80U,T. â¦\83G,Lâ¦\84 â\8a¢ â\93\9dU.T :[a,h] X2 →
- â\88§â\88§ â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U & â¦\83G,Lâ¦\84 â\8a¢ U â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #G #L #X2 #U #T #H
+lemma nta_inv_cast_sn (h) (a) (G) (L) (X2):
+ â\88\80U,T. â\9dªG,Lâ\9d« â\8a¢ â\93\9dU.T :[h,a] X2 →
+ â\88§â\88§ â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U & â\9dªG,Lâ\9d« â\8a¢ U â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,a].
+#h #a #G #L #X2 #U #T #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
qed-.
(* Basic_1: uses: ty3_gen_cast *)
-lemma nta_inv_cast_sn_old (a) (h) (G) (L) (X2):
- â\88\80T0,T1. â¦\83G,Lâ¦\84 â\8a¢ â\93\9dT1.T0 :[a,h] X2 →
- â\88\83â\88\83T2. â¦\83G,Lâ¦\84 â\8a¢ T0 :[a,h] T1 & â¦\83G,Lâ¦\84 â\8a¢ T1 :[a,h] T2 & â¦\83G,Lâ¦\84 â\8a¢ â\93\9dT2.T1 â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #G #L #X2 #T0 #T1 #H
+lemma nta_inv_cast_sn_old (h) (a) (G) (L) (X2):
+ â\88\80T0,T1. â\9dªG,Lâ\9d« â\8a¢ â\93\9dT1.T0 :[h,a] X2 →
+ â\88\83â\88\83T2. â\9dªG,Lâ\9d« â\8a¢ T0 :[h,a] T1 & â\9dªG,Lâ\9d« â\8a¢ T1 :[h,a] T2 & â\9dªG,Lâ\9d« â\8a¢ â\93\9dT2.T1 â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,a].
+#h #a #G #L #X2 #T0 #T1 #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
qed-.
(* Basic_1: uses: ty3_gen_lift *)
-(* Note: "â¦\83G, Lâ¦\84 â\8a¢ U2 â¬\8c*[h] X2" can be "â¦\83G, Lâ¦\84 ⊢ X2 ➡*[h] U2" *)
-lemma nta_inv_lifts_sn (a) (h) (G):
- â\88\80L,T2,X2. â¦\83G,Lâ¦\84 â\8a¢ T2 :[a,h] X2 →
- â\88\80b,f,K. â¬\87*[b,f] L â\89\98 K â\86\92 â\88\80T1. â¬\86*[f] T1 ≘ T2 →
- â\88\83â\88\83U1,U2. â¦\83G,Kâ¦\84 â\8a¢ T1 :[a,h] U1 & â¬\86*[f] U1 â\89\98 U2 & â¦\83G,Lâ¦\84 â\8a¢ U2 â¬\8c*[h] X2 & â¦\83G,Lâ¦\84 â\8a¢ X2 ![a,h].
-#a #h #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
+(* Note: "â\9dªG, Lâ\9d« â\8a¢ U2 â¬\8c*[h] X2" can be "â\9dªG, Lâ\9d« ⊢ X2 ➡*[h] U2" *)
+lemma nta_inv_lifts_sn (h) (a) (G):
+ â\88\80L,T2,X2. â\9dªG,Lâ\9d« â\8a¢ T2 :[h,a] X2 →
+ â\88\80b,f,K. â\87©*[b,f] L â\89\98 K â\86\92 â\88\80T1. â\87§*[f] T1 ≘ T2 →
+ â\88\83â\88\83U1,U2. â\9dªG,Kâ\9d« â\8a¢ T1 :[h,a] U1 & â\87§*[f] U1 â\89\98 U2 & â\9dªG,Lâ\9d« â\8a¢ U2 â¬\8c*[h] X2 & â\9dªG,Lâ\9d« â\8a¢ X2 ![h,a].
+#h #a #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
elim (cnv_inv_cast … H) -H #U2 #HX2 #HT2 #HXU2 #HTU2
lapply (cnv_inv_lifts … HT2 … HLK … HT12) -HT2 #HT1
elim (cpms_inv_lifts_sn … HTU2 … HLK … HT12) -T2 -HLK #U1 #HU12 #HTU1
(* Forward lemmas based on preservation *************************************)
(* Basic_1: was: ty3_unique *)
-theorem nta_mono (a) (h) (G) (L) (T):
- â\88\80U1. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U1 â\86\92 â\88\80U2. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U2 â\86\92 â¦\83G,Lâ¦\84 ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #H2
+theorem nta_mono (h) (a) (G) (L) (T):
+ â\88\80U1. â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U1 â\86\92 â\88\80U2. â\9dªG,Lâ\9d« â\8a¢ T :[h,a] U2 â\86\92 â\9dªG,Lâ\9d« ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T #U1 #H1 #U2 #H2
elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
lapply (cnv_cpms_conf_eq … HT … HTX1 … HTX2) -T #HX12
(* Advanced properties ******************************************************)
(* Basic_1: uses: ty3_sconv_pc3 *)
-lemma nta_cpcs_bi (a) (h) (G) (L):
- â\88\80T1,U1. â¦\83G,Lâ¦\84 â\8a¢ T1 :[a,h] U1 â\86\92 â\88\80T2,U2. â¦\83G,Lâ¦\84 â\8a¢ T2 :[a,h] U2 →
- â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\8c*[h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+lemma nta_cpcs_bi (h) (a) (G) (L):
+ â\88\80T1,U1. â\9dªG,Lâ\9d« â\8a¢ T1 :[h,a] U1 â\86\92 â\88\80T2,U2. â\9dªG,Lâ\9d« â\8a¢ T2 :[h,a] U2 →
+ â\9dªG,Lâ\9d« â\8a¢ T1 â¬\8c*[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
/3 width=6 by nta_mono, nta_cprs_conf/
qed-.
projects / helm.git / blobdiff