(* Advanced properties ******************************************************)
lemma cpxs_delta (G) (K):
- â\88\80I,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈* V2 →
- â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈* W2.
+ â\88\80I,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈* V2 →
+ â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\91[I]V1â\9d© ⊢ #0 ⬈* W2.
#G #K #I #V1 #V2 #H @(cpxs_ind … H) -V2
[ /3 width=3 by cpx_cpxs, cpx_delta/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
qed.
lemma cpxs_lref (G) (K):
- â\88\80I,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈* T →
- â\88\80U. â\87§[1] T â\89\98 U â\86\92 â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈* U.
+ â\88\80I,T,i. â\9d¨G,Kâ\9d© ⊢ #i ⬈* 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© ⊢ #↑i ⬈* U.
#G #K #I #T #i #H @(cpxs_ind … H) -T
[ /3 width=3 by cpx_cpxs, cpx_lref/
| #T0 #T #_ #HT2 #IH #U #HTU
(* Basic_2A1: was: cpxs_delta *)
lemma cpxs_delta_drops (G) (L):
- â\88\80I,K,V1,V2,i. â\87©[i] L â\89\98 K.â\93\91[I]V1 â\86\92 â\9dªG,Kâ\9d« ⊢ V1 ⬈* V2 →
- â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬈* W2.
+ â\88\80I,K,V1,V2,i. â\87©[i] L â\89\98 K.â\93\91[I]V1 â\86\92 â\9d¨G,Kâ\9d© ⊢ V1 ⬈* V2 →
+ â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬈* W2.
#G #L #I #K #V1 #V2 #i #HLK #H @(cpxs_ind … H) -V2
[ /3 width=7 by cpx_cpxs, cpx_delta_drops/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
(* Advanced inversion lemmas ************************************************)
lemma cpxs_inv_zero1 (G) (L):
- â\88\80T2. â\9dªG,Lâ\9d« ⊢ #0 ⬈* T2 →
+ â\88\80T2. â\9d¨G,Lâ\9d© ⊢ #0 ⬈* T2 →
∨∨ T2 = #0
- | â\88\83â\88\83I,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈* V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
+ | â\88\83â\88\83I,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈* V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
#G #L #T2 #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct
qed-.
lemma cpxs_inv_lref1 (G) (L):
- â\88\80T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ⬈* T2 →
+ â\88\80T2,i. â\9d¨G,Lâ\9d© ⊢ #↑i ⬈* T2 →
∨∨ T2 = #(↑i)
- | â\88\83â\88\83I,K,T. â\9dªG,Kâ\9d« ⊢ #i ⬈* T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+ | â\88\83â\88\83I,K,T. â\9d¨G,Kâ\9d© ⊢ #i ⬈* T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
#G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct
(* Basic_2A1: was: cpxs_inv_lref1 *)
lemma cpxs_inv_lref1_drops (G) (L):
- â\88\80T2,i. â\9dªG,Lâ\9d« ⊢ #i ⬈* T2 →
+ â\88\80T2,i. â\9d¨G,Lâ\9d© ⊢ #i ⬈* T2 →
∨∨ T2 = #i
- | â\88\83â\88\83I,K,V1,T1. â\87©[i] L â\89\98 K.â\93\91[I]V1 & â\9dªG,Kâ\9d« ⊢ V1 ⬈* T1 & ⇧[↑i] T1 ≘ T2.
+ | â\88\83â\88\83I,K,V1,T1. â\87©[i] L â\89\98 K.â\93\91[I]V1 & â\9d¨G,Kâ\9d© ⊢ V1 ⬈* T1 & ⇧[↑i] T1 ≘ T2.
#G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct