(* Advanced properties ******************************************************)
lemma feqx_dec (G1) (G2) (L1) (L2) (T1) (T2):
- Decidable (â\9dªG1,L1,T1â\9d« â\89\85 â\9dªG2,L2,T2â\9d«).
+ Decidable (â\9d¨G1,L1,T1â\9d© â\89\85 â\9d¨G2,L2,T2â\9d©).
/3 width=1 by feqg_dec, sfull_dec/ qed-.
(*
lemma feqx_sym: tri_symmetric … feqx.
/4 width=5 by feqx_intro_sn, reqx_trans, teqx_reqx_div, teqx_trans/
qed-.
-theorem feqx_canc_sn: â\88\80G,G1,L,L1,T,T1. â\9dªG,L,Tâ\9d« â\89\9b â\9dªG1,L1,T1â\9d«→
- â\88\80G2,L2,T2. â\9dªG,L,Tâ\9d« â\89\9b â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89\9b â\9dªG2,L2,T2â\9d«.
+theorem feqx_canc_sn: â\88\80G,G1,L,L1,T,T1. â\9d¨G,L,Tâ\9d© â\89\9b â\9d¨G1,L1,T1â\9d©→
+ â\88\80G2,L2,T2. â\9d¨G,L,Tâ\9d© â\89\9b â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89\9b â\9d¨G2,L2,T2â\9d©.
/3 width=5 by feqx_trans, feqx_sym/ qed-.
-theorem feqx_canc_dx: â\88\80G1,G,L1,L,T1,T. â\9dªG1,L1,T1â\9d« â\89\9b â\9dªG,L,Tâ\9d« →
- â\88\80G2,L2,T2. â\9dªG2,L2,T2â\9d« â\89\9b â\9dªG,L,Tâ\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89\9b â\9dªG2,L2,T2â\9d«.
+theorem feqx_canc_dx: â\88\80G1,G,L1,L,T1,T. â\9d¨G1,L1,T1â\9d© â\89\9b â\9d¨G,L,Tâ\9d© →
+ â\88\80G2,L2,T2. â\9d¨G2,L2,T2â\9d© â\89\9b â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89\9b â\9d¨G2,L2,T2â\9d©.
/3 width=5 by feqx_trans, feqx_sym/ qed-.
(* Main inversion lemmas with degree-based equivalence on terms *************)
-theorem feqx_tneqx_repl_dx: â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â\89\9b â\9dªG2,L2,T2â\9d« →
- â\88\80U1,U2. â\9dªG1,L1,U1â\9d« â\89\9b â\9dªG2,L2,U2â\9d« →
+theorem feqx_tneqx_repl_dx: â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â\89\9b â\9d¨G2,L2,T2â\9d© →
+ â\88\80U1,U2. â\9d¨G1,L1,U1â\9d© â\89\9b â\9d¨G2,L2,U2â\9d© →
(T2 ≛ U2 → ⊥) → (T1 ≛ U1 → ⊥).
#G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
elim (feqx_inv_gen_sn … HT) -HT #_ #_ #HT