definition f_dropable_sn:
predicate (relation3 lenv term term) ≝ λR.
- â\88\80b,f,L1,K1. â\87©*[b,f] L1 â\89\98 K1 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
+ â\88\80b,f,L1,K1. â\87©*[b,f] L1 â\89\98 K1 â\86\92 ð\9d\90\94â\9d¨fâ\9d© →
∀L2,U. L1 ⪤[R,U] L2 → ∀T. ⇧*[f] T ≘ U →
∃∃K2. K1 ⪤[R,T] K2 & ⇩*[b,f] L2 ≘ K2.
definition f_dropable_dx:
predicate (relation3 lenv term term) ≝ λR.
∀L1,L2,U. L1 ⪤[R,U] L2 →
- â\88\80b,f,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â\9dªfâ\9d« → ∀T. ⇧*[f] T ≘ U →
+ â\88\80b,f,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â\9d¨fâ\9d© → ∀T. ⇧*[f] T ≘ U →
∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[R,T] K2.
definition f_transitive_next:
relation3 … ≝ λR1,R2,R3.
- â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f →
+ â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â\9d¨Tâ\9d© ≘ f →
∀g,I,K,i. ⇩[i] L ≘ K.ⓘ[I] → ↑g = ⫰*[i] f →
R_pw_transitive_sex (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
definition f_confluent1_next: relation2 … ≝ λR1,R2.
- â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f →
+ â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â\9d¨Tâ\9d© ≘ f →
∀g,I,K,i. ⇩[i] L ≘ K.ⓘ[I] → ↑g = ⫰*[i] f →
R_pw_confluent1_sex (cext2 R1) (cext2 R1) (cext2 R2) cfull g K I.
(* Basic_2A1: uses: llpx_sn_inv_lift_O *)
lemma rex_inv_lifts_bi (R):
- â\88\80L1,L2,U. L1 ⪤[R,U] L2 â\86\92 â\88\80b,f. ð\9d\90\94â\9dªfâ\9d« →
+ â\88\80L1,L2,U. L1 ⪤[R,U] L2 â\86\92 â\88\80b,f. ð\9d\90\94â\9d¨fâ\9d© →
∀K1,K2. ⇩*[b,f] L1 ≘ K1 → ⇩*[b,f] L2 ≘ K2 →
∀T. ⇧*[f] T ≘ U → K1 ⪤[R,T] K2.
#R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
lemma rex_inv_lref_unit_sn (R):
∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1. ⇩[i] L1 ≘ K1.ⓤ[I] →
- â\88\83â\88\83f,K2. â\87©[i] L2 â\89\98 K2.â\93¤[I] & K1 ⪤[cext2 R,cfull,f] K2 & ð\9d\90\88â\9dªfâ\9d«.
+ â\88\83â\88\83f,K2. â\87©[i] L2 â\89\98 K2.â\93¤[I] & K1 ⪤[cext2 R,cfull,f] K2 & ð\9d\90\88â\9d¨fâ\9d©.
#R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (rex_inv_zero_unit_sn … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
lemma rex_inv_lref_unit_dx (R):
∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2. ⇩[i] L2 ≘ K2.ⓤ[I] →
- â\88\83â\88\83f,K1. â\87©[i] L1 â\89\98 K1.â\93¤[I] & K1 ⪤[cext2 R,cfull,f] K2 & ð\9d\90\88â\9dªfâ\9d«.
+ â\88\83â\88\83f,K1. â\87©[i] L1 â\89\98 K1.â\93¤[I] & K1 ⪤[cext2 R,cfull,f] K2 & ð\9d\90\88â\9d¨fâ\9d©.
#R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (rex_inv_zero_unit_dx … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/