(* Properties with generic slicing for local environments *******************)
-lemma cpms_lifts_sn: ∀n,h,G. d_liftable2_sn … lifts (λL. cpms h G L n).
+lemma cpms_lifts_sn: ∀h,n,G. d_liftable2_sn … lifts (λL. cpms h G L n).
/3 width=6 by d2_liftable_sn_ltc, cpm_lifts_sn/ qed-.
(* Basic_2A1: uses: scpds_lift *)
(* Basic_2A1: includes: cprs_lift *)
(* Basic_1: includes: pr3_lift *)
-lemma cpms_lifts_bi: ∀n,h,G. d_liftable2_bi … lifts (λL. cpms h G L n).
-#n #h #G @d_liftable2_sn_bi
+lemma cpms_lifts_bi: ∀h,n,G. d_liftable2_bi … lifts (λL. cpms h G L n).
+#h #n #G @d_liftable2_sn_bi
/2 width=6 by cpms_lifts_sn, lifts_mono/
qed-.
(* Basic_2A1: uses: scpds_inv_lift1 *)
(* Basic_2A1: includes: cprs_inv_lift1 *)
(* Basic_1: includes: pr3_gen_lift *)
-lemma cpms_inv_lifts_sn: ∀n,h,G. d_deliftable2_sn … lifts (λL. cpms h G L n).
+lemma cpms_inv_lifts_sn: ∀h,n,G. d_deliftable2_sn … lifts (λL. cpms h G L n).
/3 width=6 by d2_deliftable_sn_ltc, cpm_inv_lifts_sn/ qed-.
-lemma cpms_inv_lifts_bi: ∀n,h,G. d_deliftable2_bi … lifts (λL. cpms h G L n).
-#n #h #G @d_deliftable2_sn_bi
+lemma cpms_inv_lifts_bi: ∀h,n,G. d_deliftable2_bi … lifts (λL. cpms h G L n).
+#h #n #G @d_deliftable2_sn_bi
/2 width=6 by cpms_inv_lifts_sn, lifts_inj/
qed-.
(* Advanced properties ******************************************************)
-lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
- â\88\80W2. â¬\86*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\93V1â¦\84 â\8a¢ #0 â\9e¡*[n,h] W2.
-#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
+lemma cpms_delta (h) (n) (G): ∀K,V1,V2. ❨G,K❩ ⊢ V1 ➡*[h,n] V2 →
+ â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\93V1â\9d© â\8a¢ #0 â\9e¡*[h,n] W2.
+#h #n #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_delta/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
- elim (lifts_total V (ð\9d\90\94â\9d´1â\9dµ)) #W #HVW
+ elim (lifts_total V (ð\9d\90\94â\9d¨1â\9d©)) #W #HVW
/5 width=11 by cpms_step_dx, cpm_lifts_bi, drops_refl, drops_drop/
]
qed.
-lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
- â\88\80W2. â¬\86*[1] V2 â\89\98 W2 â\86\92 â¦\83G,K.â\93\9bV1â¦\84 â\8a¢ #0 â\9e¡*[â\86\91n,h] W2.
-#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
+lemma cpms_ell (h) (n) (G): ∀K,V1,V2. ❨G,K❩ ⊢ V1 ➡*[h,n] V2 →
+ â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\9bV1â\9d© â\8a¢ #0 â\9e¡*[h,â\86\91n] W2.
+#h #n #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_ell/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
- elim (lifts_total V (ð\9d\90\94â\9d´1â\9dµ)) #W #HVW >plus_S1
+ elim (lifts_total V (ð\9d\90\94â\9d¨1â\9d©)) #W #HVW >plus_S1
/5 width=11 by cpms_step_dx, cpm_lifts_bi, drops_refl, drops_drop/
]
qed.
-lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ⦃G,K⦄ ⊢ #i ➡*[n,h] T →
- â\88\80U. â¬\86*[1] T â\89\98 U â\86\92 â¦\83G,K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i â\9e¡*[n,h] U.
-#n #h #I #G #K #T #i #H @(cpms_ind_dx … H) -T
+lemma cpms_lref (h) (n) (I) (G): ∀K,T,i. ❨G,K❩ ⊢ #i ➡*[h,n] T →
+ â\88\80U. â\87§[1] T â\89\98 U â\86\92 â\9d¨G,K.â\93\98[I]â\9d© â\8a¢ #â\86\91i â\9e¡*[h,n] U.
+#h #n #I #G #K #T #i #H @(cpms_ind_dx … H) -T
[ /3 width=3 by cpm_cpms, cpm_lref/
| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
- elim (lifts_total T (ð\9d\90\94â\9d´1â\9dµ)) #U #TU
+ elim (lifts_total T (ð\9d\90\94â\9d¨1â\9d©)) #U #TU
/5 width=11 by cpms_step_dx, cpm_lifts_bi, drops_refl, drops_drop/
]
qed.
-lemma cpms_cast_sn (n) (h) (G) (L):
- â\88\80U1,U2. â¦\83G,Lâ¦\84 â\8a¢ U1 â\9e¡*[n,h] U2 →
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[n,h] T2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\9dU1.T1 â\9e¡*[n,h] ⓝU2.T2.
-#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
+lemma cpms_cast_sn (h) (n) (G) (L):
+ â\88\80U1,U2. â\9d¨G,Lâ\9d© â\8a¢ U1 â\9e¡*[h,n] U2 →
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 →
+ â\9d¨G,Lâ\9d© â\8a¢ â\93\9dU1.T1 â\9e¡*[h,n] ⓝU2.T2.
+#h #n #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=3 by cpm_cpms, cpm_cast/
| #n1 #n2 #U1 #U #HU1 #_ #IH #T1 #T2 #H
elim (cpm_fwd_plus … H) -H #T #HT1 #HT2
(* Note: apparently this was missing in basic_1 *)
(* Basic_2A1: uses: cprs_delta *)
-lemma cpms_delta_drops (n) (h) (G):
- â\88\80L,K,V,i. â¬\87*[i] L ≘ K.ⓓV →
- â\88\80V2. â¦\83G,Kâ¦\84 â\8a¢ V â\9e¡*[n,h] V2 →
- â\88\80W2. â¬\86*[â\86\91i] V2 â\89\98 W2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ #i â\9e¡*[n,h] W2.
-#n #h #G #L #K #V #i #HLK #V2 #H @(cpms_ind_dx … H) -V2
+lemma cpms_delta_drops (h) (n) (G):
+ â\88\80L,K,V,i. â\87©[i] L ≘ K.ⓓV →
+ â\88\80V2. â\9d¨G,Kâ\9d© â\8a¢ V â\9e¡*[h,n] V2 →
+ â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ #i â\9e¡*[h,n] W2.
+#h #n #G #L #K #V #i #HLK #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=6 by cpm_cpms, cpm_delta_drops/
| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
- elim (lifts_total V1 (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #W1 #HVW1
+ elim (lifts_total V1 (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #W1 #HVW1
/3 width=11 by cpm_lifts_bi, cpms_step_dx/
]
qed.
-lemma cpms_ell_drops (n) (h) (G):
- â\88\80L,K,W,i. â¬\87*[i] L ≘ K.ⓛW →
- â\88\80W2. â¦\83G,Kâ¦\84 â\8a¢ W â\9e¡*[n,h] W2 →
- â\88\80V2. â¬\86*[â\86\91i] W2 â\89\98 V2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ #i â\9e¡*[â\86\91n,h] V2.
-#n #h #G #L #K #W #i #HLK #W2 #H @(cpms_ind_dx … H) -W2
+lemma cpms_ell_drops (h) (n) (G):
+ â\88\80L,K,W,i. â\87©[i] L ≘ K.ⓛW →
+ â\88\80W2. â\9d¨G,Kâ\9d© â\8a¢ W â\9e¡*[h,n] W2 →
+ â\88\80V2. â\87§[â\86\91i] W2 â\89\98 V2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ #i â\9e¡*[h,â\86\91n] V2.
+#h #n #G #L #K #W #i #HLK #W2 #H @(cpms_ind_dx … H) -W2
[ /3 width=6 by cpm_cpms, cpm_ell_drops/
| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
- elim (lifts_total W1 (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #V1 #HWV1 >plus_S1
+ elim (lifts_total W1 (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #V1 #HWV1 >plus_S1
/3 width=11 by cpm_lifts_bi, cpms_step_dx/
]
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpms_inv_lref1_drops (n) (h) (G):
- â\88\80L,T2,i. â¦\83G,Lâ¦\84 â\8a¢ #i â\9e¡*[n,h] T2 →
+lemma cpms_inv_lref1_drops (h) (n) (G):
+ â\88\80L,T2,i. â\9d¨G,Lâ\9d© â\8a¢ #i â\9e¡*[h,n] T2 →
∨∨ ∧∧ T2 = #i & n = 0
- | â\88\83â\88\83K,V,V2. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G,Kâ¦\84 â\8a¢ V â\9e¡*[n,h] V2 &
- â¬\86*[↑i] V2 ≘ T2
- | â\88\83â\88\83m,K,V,V2. â¬\87*[i] L â\89\98 K.â\93\9bV & â¦\83G,Kâ¦\84 â\8a¢ V â\9e¡*[m,h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & n = ↑m.
-#n #h #G #L #T2 #i #H @(cpms_ind_dx … H) -T2
+ | â\88\83â\88\83K,V,V2. â\87©[i] L â\89\98 K.â\93\93V & â\9d¨G,Kâ\9d© â\8a¢ V â\9e¡*[h,n] V2 &
+ â\87§[↑i] V2 ≘ T2
+ | â\88\83â\88\83m,K,V,V2. â\87©[i] L â\89\98 K.â\93\9bV & â\9d¨G,Kâ\9d© â\8a¢ V â\9e¡*[h,m] V2 &
+ â\87§[↑i] V2 ≘ T2 & n = ↑m.
+#h #n #G #L #T2 #i #H @(cpms_ind_dx … H) -T2
[ /3 width=1 by or3_intro0, conj/
| #n1 #n2 #T #T2 #_ #IH #HT2 cases IH -IH *
[ #H1 #H2 destruct
]
qed-.
-lemma cpms_inv_delta_sn (n) (h) (G) (K) (V):
- â\88\80T2. â¦\83G,K.â\93\93Vâ¦\84 â\8a¢ #0 â\9e¡*[n,h] T2 →
+lemma cpms_inv_delta_sn (h) (n) (G) (K) (V):
+ â\88\80T2. â\9d¨G,K.â\93\93Vâ\9d© â\8a¢ #0 â\9e¡*[h,n] T2 →
∨∨ ∧∧ T2 = #0 & n = 0
- | â\88\83â\88\83V2. â¦\83G,Kâ¦\84 â\8a¢ V â\9e¡*[n,h] V2 & â¬\86*[1] V2 ≘ T2.
-#n #h #G #K #V #T2 #H
+ | â\88\83â\88\83V2. â\9d¨G,Kâ\9d© â\8a¢ V â\9e¡*[h,n] V2 & â\87§[1] V2 ≘ T2.
+#h #n #G #K #V #T2 #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/
| #Y #X #V2 #H #HV2 #HVT2
]
qed-.
-lemma cpms_inv_ell_sn (n) (h) (G) (K) (V):
- â\88\80T2. â¦\83G,K.â\93\9bVâ¦\84 â\8a¢ #0 â\9e¡*[n,h] T2 →
+lemma cpms_inv_ell_sn (h) (n) (G) (K) (V):
+ â\88\80T2. â\9d¨G,K.â\93\9bVâ\9d© â\8a¢ #0 â\9e¡*[h,n] T2 →
∨∨ ∧∧ T2 = #0 & n = 0
- | â\88\83â\88\83m,V2. â¦\83G,Kâ¦\84 â\8a¢ V â\9e¡*[m,h] V2 & â¬\86*[1] V2 ≘ T2 & n = ↑m.
-#n #h #G #K #V #T2 #H
+ | â\88\83â\88\83m,V2. â\9d¨G,Kâ\9d© â\8a¢ V â\9e¡*[h,m] V2 & â\87§[1] V2 ≘ T2 & n = ↑m.
+#h #n #G #K #V #T2 #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/
| #Y #X #V2 #H #HV2 #HVT2
]
qed-.
-lemma cpms_inv_lref_sn (n) (h) (G) (I) (K):
- â\88\80U2,i. â¦\83G,K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i â\9e¡*[n,h] U2 →
+lemma cpms_inv_lref_sn (h) (n) (G) (I) (K):
+ â\88\80U2,i. â\9d¨G,K.â\93\98[I]â\9d© â\8a¢ #â\86\91i â\9e¡*[h,n] U2 →
∨∨ ∧∧ U2 = #↑i & n = 0
- | â\88\83â\88\83T2. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡*[n,h] T2 & â¬\86*[1] T2 ≘ U2.
-#n #h #G #I #K #U2 #i #H
+ | â\88\83â\88\83T2. â\9d¨G,Kâ\9d© â\8a¢ #i â\9e¡*[h,n] T2 & â\87§[1] T2 ≘ U2.
+#h #n #G #I #K #U2 #i #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/
| #L #V #V2 #H #HV2 #HVU2
lapply (drops_inv_drop1 … H) -H #HLK
- elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ)) -HVU2
+ elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©)) -HVU2
[| // ] #T2 #HVT2 #HTU2
/4 width=6 by cpms_delta_drops, ex2_intro, or_intror/
| #m #L #V #V2 #H #HV2 #HVU2 #H0 destruct
lapply (drops_inv_drop1 … H) -H #HLK
- elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ)) -HVU2
+ elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©)) -HVU2
[| // ] #T2 #HVT2 #HTU2
/4 width=6 by cpms_ell_drops, ex2_intro, or_intror/
]
qed-.
-fact cpms_inv_succ_sn (n) (h) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[â\86\91n,h] T2 →
- â\88\83â\88\83T. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[1,h] T & â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n,h] T2.
-#n #h #G #L #T1 #T2
+fact cpms_inv_succ_sn (h) (n) (G) (L):
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,â\86\91n] T2 →
+ â\88\83â\88\83T. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,1] T & â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n] T2.
+#h #n #G #L #T1 #T2
@(insert_eq_0 … (↑n)) #m #H
@(cpms_ind_sn … H) -T1 -m
[ #H destruct