(* *)
(**************************************************************************)
-include "ground_2/xoa/ex_5_1.ma".
-include "ground_2/xoa/ex_8_5.ma".
-include "ground_2/xoa/ex_9_3.ma".
+include "ground/xoa/ex_5_1.ma".
+include "ground/xoa/ex_8_5.ma".
+include "ground/xoa/ex_9_3.ma".
include "basic_2/rt_transition/cpm_teqx.ma".
-include "basic_2/rt_computation/fpbg_fqup.ma".
+include "basic_2/rt_computation/fpbg_cpm.ma".
include "basic_2/dynamic/cnv_fsb.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Inversion lemmas with restricted rt-transition for terms *****************)
lemma cnv_cpr_teqx_fwd_refl (h) (a) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h] T2 â\86\92 T1 â\89\9b T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ![h,a] → T1 = T2.
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,0] T2 â\86\92 T1 â\89\85 T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ![h,a] → T1 = T2.
#h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
- lapply (teqx_inv_lref1 … H1) -H1 #H destruct //
+ lapply (teqg_inv_lref1 … H1) -H1 #H destruct //
| #I #G #K #T2 #X2 #i #_ #_ #_ #H1 #_ -a -I -G -K -T2
- lapply (teqx_inv_lref1 … H1) -H1 #H destruct //
+ lapply (teqg_inv_lref1 … H1) -H1 #H destruct //
| #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
elim (teqx_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
elim (cnv_inv_bind … H2) -H2 #HV1 #HT1
/3 width=3 by eq_f2/
| #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
elim (cnv_fpbg_refl_false … H2) -a
- @(fpbg_teqx_div … H1) -H1
- /3 width=9 by cpm_tneqx_cpm_fpbg, cpm_zeta, teqx_lifts_inv_pair_sn/
+ @(fpbg_teqg_div … H1) -H1
+ /3 width=9 by cpm_tneqx_cpm_fpbg, cpm_zeta, teqg_lifts_inv_pair_sn/
| #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
elim (cnv_fpbg_refl_false … H2) -a
- @(fpbg_teqx_div … H1) -H1
- /3 width=6 by cpm_tneqx_cpm_fpbg, cpm_eps, teqx_inv_pair_xy_y/
+ @(fpbg_teqg_div … H1) -H1
+ /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_eps, teqg_inv_pair_xy_y/
| #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
elim (teqx_inv_pair … H1) -H1 #H #_ #_ destruct
| #p #G #L #V1 #V2 #X2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H1 #_
qed-.
lemma cpm_teqx_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
- â\88\80V,T1. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V.T1 ![h,a] →
- â\88\80X. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V.T1 â\9e¡[n,h] X â\86\92 â\93\91[p,I]V.T1 â\89\9b X →
- â\88\83â\88\83T2. â\9dªG,Lâ\9d« â\8a¢ V ![h,a] & â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 ![h,a] & â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 & T1 â\89\9b T2 & X = ⓑ[p,I]V.T2.
+ â\88\80V,T1. â\9d¨G,Lâ\9d© ⊢ ⓑ[p,I]V.T1 ![h,a] →
+ â\88\80X. â\9d¨G,Lâ\9d© â\8a¢ â\93\91[p,I]V.T1 â\9e¡[h,n] X â\86\92 â\93\91[p,I]V.T1 â\89\85 X →
+ â\88\83â\88\83T2. â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] & â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 ![h,a] & â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 & T1 â\89\85 T2 & X = ⓑ[p,I]V.T2.
#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
/2 width=4 by ex5_intro/
| #X1 #HXT1 #HX1 #H1 #H destruct
elim (cnv_fpbg_refl_false … H0) -a
- @(fpbg_teqx_div … H2) -H2
+ @(fpbg_teqg_div … H2) -H2
/3 width=9 by cpm_tneqx_cpm_fpbg, cpm_zeta, teqx_lifts_inv_pair_sn/
]
qed-.
lemma cpm_teqx_inv_appl_sn (h) (a) (n) (G) (L):
- â\88\80V,T1. â\9dªG,Lâ\9d« ⊢ ⓐV.T1 ![h,a] →
- â\88\80X. â\9dªG,Lâ\9d« â\8a¢ â\93\90V.T1 â\9e¡[n,h] X â\86\92 â\93\90V.T1 â\89\9b X →
- â\88\83â\88\83m,q,W,U1,T2. ad a m & â\9dªG,Lâ\9d« â\8a¢ V ![h,a] & â\9dªG,Lâ\9d« â\8a¢ V â\9e¡*[1,h] W & â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[m,h] ⓛ[q]W.U1
- & â\9dªG,Lâ\9d«â\8a¢ T1 ![h,a] & â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 & T1 â\89\9b T2 & X = ⓐV.T2.
+ â\88\80V,T1. â\9d¨G,Lâ\9d© ⊢ ⓐV.T1 ![h,a] →
+ â\88\80X. â\9d¨G,Lâ\9d© â\8a¢ â\93\90V.T1 â\9e¡[h,n] X â\86\92 â\93\90V.T1 â\89\85 X →
+ â\88\83â\88\83m,q,W,U1,T2. ad a m & â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] & â\9d¨G,Lâ\9d© â\8a¢ V â\9e¡*[h,1] W & â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,m] ⓛ[q]W.U1
+ & â\9d¨G,Lâ\9d©â\8a¢ T1 ![h,a] & â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 & T1 â\89\85 T2 & X = ⓐV.T2.
#h #a #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
qed-.
lemma cpm_teqx_inv_cast_sn (h) (a) (n) (G) (L):
- â\88\80U1,T1. â\9dªG,Lâ\9d« ⊢ ⓝU1.T1 ![h,a] →
- â\88\80X. â\9dªG,Lâ\9d« â\8a¢ â\93\9dU1.T1 â\9e¡[n,h] X â\86\92 â\93\9dU1.T1 â\89\9b X →
- â\88\83â\88\83U0,U2,T2. â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡*[h] U0 & â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[1,h] U0
- & â\9dªG,Lâ\9d« â\8a¢ U1 ![h,a] & â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡[n,h] U2 & U1 â\89\9b U2
- & â\9dªG,Lâ\9d« â\8a¢ T1 ![h,a] & â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 & T1 â\89\9b T2 & X = ⓝU2.T2.
+ â\88\80U1,T1. â\9d¨G,Lâ\9d© ⊢ ⓝU1.T1 ![h,a] →
+ â\88\80X. â\9d¨G,Lâ\9d© â\8a¢ â\93\9dU1.T1 â\9e¡[h,n] X â\86\92 â\93\9dU1.T1 â\89\85 X →
+ â\88\83â\88\83U0,U2,T2. â\9d¨G,Lâ\9d© â\8a¢ U1 â\9e¡*[h,0] U0 & â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,1] U0
+ & â\9d¨G,Lâ\9d© â\8a¢ U1 ![h,a] & â\9d¨G,Lâ\9d© â\8a¢ U1 â\9e¡[h,n] U2 & U1 â\89\85 U2
+ & â\9d¨G,Lâ\9d© â\8a¢ T1 ![h,a] & â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 & T1 â\89\85 T2 & X = ⓝU2.T2.
#h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
/2 width=7 by ex9_3_intro/
| #HT1X
elim (cnv_fpbg_refl_false … H0) -a
- @(fpbg_teqx_div … H2) -H2
- /3 width=6 by cpm_tneqx_cpm_fpbg, cpm_eps, teqx_inv_pair_xy_y/
+ @(fpbg_teqg_div … H2) -H2
+ /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_eps, teqg_inv_pair_xy_y/
| #m #HU1X #H destruct
elim (cnv_fpbg_refl_false … H0) -a
- @(fpbg_teqx_div … H2) -H2
- /3 width=6 by cpm_tneqx_cpm_fpbg, cpm_ee, teqx_inv_pair_xy_x/
+ @(fpbg_teqg_div … H2) -H2
+ /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_ee, teqg_inv_pair_xy_x/
]
qed-.
lemma cpm_teqx_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
- â\88\80X. â\9dªG,Lâ\9d« ⊢ X ![h,a] →
- â\88\80V,T2. â\9dªG,Lâ\9d« â\8a¢ X â\9e¡[n,h] â\93\91[p,I]V.T2 â\86\92 X â\89\9b ⓑ[p,I]V.T2 →
- â\88\83â\88\83T1. â\9dªG,Lâ\9d« â\8a¢ V ![h,a] & â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 ![h,a] & â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 & T1 â\89\9b T2 & X = ⓑ[p,I]V.T1.
+ â\88\80X. â\9d¨G,Lâ\9d© ⊢ X ![h,a] →
+ â\88\80V,T2. â\9d¨G,Lâ\9d© â\8a¢ X â\9e¡[h,n] â\93\91[p,I]V.T2 â\86\92 X â\89\85 ⓑ[p,I]V.T2 →
+ â\88\83â\88\83T1. â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] & â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 ![h,a] & â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 & T1 â\89\85 T2 & X = ⓑ[p,I]V.T1.
#h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (teqx_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
lemma cpm_teqx_ind (h) (a) (n) (G) (Q:relation3 …):
(∀I,L. n = 0 → Q L (⓪[I]) (⓪[I])) →
(∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
- (â\88\80p,I,L,V,T1. â\9dªG,Lâ\9d«â\8a¢ V![h,a] â\86\92 â\9dªG,L.â\93\91[I]Vâ\9d«⊢T1![h,a] →
- â\88\80T2. â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 T1 â\89\9b T2 →
+ (â\88\80p,I,L,V,T1. â\9d¨G,Lâ\9d©â\8a¢ V![h,a] â\86\92 â\9d¨G,L.â\93\91[I]Vâ\9d©⊢T1![h,a] →
+ â\88\80T2. â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 T1 â\89\85 T2 →
Q (L.ⓑ[I]V) T1 T2 → Q L (ⓑ[p,I]V.T1) (ⓑ[p,I]V.T2)
) →
(∀m. ad a m →
- â\88\80L,V. â\9dªG,Lâ\9d« â\8a¢ V ![h,a] â\86\92 â\88\80W. â\9dªG,Lâ\9d« â\8a¢ V â\9e¡*[1,h] W →
- â\88\80p,T1,U1. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[m,h] â\93\9b[p]W.U1 â\86\92 â\9dªG,Lâ\9d«⊢ T1 ![h,a] →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 T1 â\89\9b T2 →
+ â\88\80L,V. â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] â\86\92 â\88\80W. â\9d¨G,Lâ\9d© â\8a¢ V â\9e¡*[h,1] W →
+ â\88\80p,T1,U1. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,m] â\93\9b[p]W.U1 â\86\92 â\9d¨G,Lâ\9d©⊢ T1 ![h,a] →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 T1 â\89\85 T2 →
Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
) →
- (â\88\80L,U0,U1,T1. â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡*[h] U0 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[1,h] U0 →
- â\88\80U2. â\9dªG,Lâ\9d« â\8a¢ U1 ![h,a] â\86\92 â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡[n,h] U2 â\86\92 U1 â\89\9b U2 →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 ![h,a] â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 T1 â\89\9b T2 →
+ (â\88\80L,U0,U1,T1. â\9d¨G,Lâ\9d© â\8a¢ U1 â\9e¡*[h,0] U0 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,1] U0 →
+ â\88\80U2. â\9d¨G,Lâ\9d© â\8a¢ U1 ![h,a] â\86\92 â\9d¨G,Lâ\9d© â\8a¢ U1 â\9e¡[h,n] U2 â\86\92 U1 â\89\85 U2 →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 ![h,a] â\86\92 â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 T1 â\89\85 T2 →
Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
) →
- â\88\80L,T1. â\9dªG,Lâ\9d« ⊢ T1 ![h,a] →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 T1 â\89\9b T2 → Q L T1 T2.
+ â\88\80L,T1. â\9d¨G,Lâ\9d© ⊢ T1 ![h,a] →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 T1 â\89\85 T2 → Q L T1 T2.
#h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
@(insert_eq_0 … G) #F
@(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
(* Advanced properties with restricted rt-transition for terms **************)
lemma cpm_teqx_free (h) (a) (n) (G) (L):
- â\88\80T1. â\9dªG,Lâ\9d« ⊢ T1 ![h,a] →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 T1 â\89\9b T2 →
- â\88\80F,K. â\9dªF,Kâ\9d« â\8a¢ T1 â\9e¡[n,h] T2.
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ T1 ![h,a] →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 T1 â\89\85 T2 →
+ â\88\80F,K. â\9d¨F,Kâ\9d© â\8a¢ T1 â\9e¡[h,n] T2.
#h #a #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_teqx_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
(* Advanced inversion lemmas with restricted rt-transition for terms ********)
lemma cpm_teqx_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
- â\88\80V,T1. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V.T1 ![h,a] →
- â\88\80X. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V.T1 â\9e¡[n,h] X â\86\92 â\93\91[p,I]V.T1 â\89\9b X →
- â\88\83â\88\83T2. â\9dªG,Lâ\9d« â\8a¢ V ![h,a] & â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ T1 ![h,a] & â\9dªG,L.â\93§â\9d« â\8a¢ T1 â\9e¡[n,h] T2 & T1 â\89\9b T2 & X = ⓑ[p,I]V.T2.
+ â\88\80V,T1. â\9d¨G,Lâ\9d© ⊢ ⓑ[p,I]V.T1 ![h,a] →
+ â\88\80X. â\9d¨G,Lâ\9d© â\8a¢ â\93\91[p,I]V.T1 â\9e¡[h,n] X â\86\92 â\93\91[p,I]V.T1 â\89\85 X →
+ â\88\83â\88\83T2. â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] & â\9d¨G,L.â\93\91[I]Vâ\9d© â\8a¢ T1 ![h,a] & â\9d¨G,L.â\93§â\9d© â\8a¢ T1 â\9e¡[h,n] T2 & T1 â\89\85 T2 & X = ⓑ[p,I]V.T2.
#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_teqx_free/