(* *)
(**************************************************************************)
-include "ground_2/lib/star.ma".
+include "ground/lib/star.ma".
include "static_2/notation/relations/suptermplus_6.ma".
include "static_2/notation/relations/suptermplus_7.ma".
include "static_2/s_transition/fqu.ma".
(* Basic properties *********************************************************)
-lemma fqu_fqup: â\88\80b,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90[b] â¦\83G2, L2, T2â¦\84 →
- â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84.
+lemma fqu_fqup: â\88\80b,G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© →
+ â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d©.
/2 width=1 by tri_inj/ qed.
lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G, L, Tâ¦\84 â\8a\90[b] â¦\83G2, L2, T2â¦\84 →
- â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84.
+ â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G,L,Tâ\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© →
+ â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d©.
/2 width=5 by tri_step/ qed.
lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- â¦\83G1, L1, T1â¦\84 â\8a\90[b] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G, L, Tâ¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84 →
- â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84.
+ â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G,L,Tâ\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© →
+ â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d©.
/2 width=5 by tri_TC_strap/ qed.
-lemma fqup_pair_sn: â\88\80b,I,G,L,V,T. â¦\83G, L, â\91¡{I}V.Tâ¦\84 â\8a\90+[b] â¦\83G, L, Vâ¦\84.
+lemma fqup_pair_sn: â\88\80b,I,G,L,V,T. â\9d¨G,L,â\91¡[I]V.Tâ\9d© â¬\82+[b] â\9d¨G,L,Vâ\9d©.
/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
-/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
+lemma fqup_bind_dx: ∀p,I,G,L,V,T. ❨G,L,ⓑ[p,I]V.T❩ ⬂+[Ⓣ] ❨G,L.ⓑ[I]V,T❩.
+/3 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-lemma fqup_clear: â\88\80p,I,G,L,V,T. â¦\83G, L, â\93\91{p,I}V.Tâ¦\84 â\8a\90+[â\92»] â¦\83G, L.â\93§, Tâ¦\84.
+lemma fqup_clear: â\88\80p,I,G,L,V,T. â\9d¨G,L,â\93\91[p,I]V.Tâ\9d© â¬\82+[â\92»] â\9d¨G,L.â\93§,Tâ\9d©.
/3 width=1 by fqu_clear, fqu_fqup/ qed.
-lemma fqup_flat_dx: â\88\80b,I,G,L,V,T. â¦\83G, L, â\93\95{I}V.Tâ¦\84 â\8a\90+[b] â¦\83G, L, Tâ¦\84.
+lemma fqup_flat_dx: â\88\80b,I,G,L,V,T. â\9d¨G,L,â\93\95[I]V.Tâ\9d© â¬\82+[b] â\9d¨G,L,Tâ\9d©.
/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-lemma fqup_flat_dx_pair_sn: â\88\80b,I1,I2,G,L,V1,V2,T. â¦\83G, L, â\93\95{I1}V1.â\91¡{I2}V2.Tâ¦\84 â\8a\90+[b] â¦\83G, L, V2â¦\84.
+lemma fqup_flat_dx_pair_sn: â\88\80b,I1,I2,G,L,V1,V2,T. â\9d¨G,L,â\93\95[I1]V1.â\91¡[I2]V2.Tâ\9d© â¬\82+[b] â\9d¨G,L,V2â\9d©.
/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
+lemma fqup_bind_dx_flat_dx: ∀p,G,I1,I2,L,V1,V2,T. ❨G,L,ⓑ[p,I1]V1.ⓕ[I2]V2.T❩ ⬂+[Ⓣ] ❨G,L.ⓑ[I1]V1,T❩.
/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
-/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
+lemma fqup_flat_dx_bind_dx: ∀p,I1,I2,G,L,V1,V2,T. ❨G,L,ⓕ[I1]V1.ⓑ[p,I2]V2.T❩ ⬂+[Ⓣ] ❨G,L.ⓑ[I2]V2,T❩.
+/3 width=5 by fqu_bind_dx, fqup_strap1/ qed.
(* Basic eliminators ********************************************************)
lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
- (â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 â\8a\90[b] â¦\83G2, L2, T2â¦\84 → Q G2 L2 T2) →
- (â\88\80G,G2,L,L2,T,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G, L, Tâ¦\84 â\8a\90[b] â¦\83G2, L2, T2â¦\84 → Q G L T → Q G2 L2 T2) →
- â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84 → Q G2 L2 T2.
+ (â\88\80G2,L2,T2. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© → Q G2 L2 T2) →
+ (â\88\80G,G2,L,L2,T,T2. â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G,L,Tâ\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© → Q G L T → Q G2 L2 T2) →
+ â\88\80G2,L2,T2. â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© → Q G2 L2 T2.
#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
qed-.
lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
- (â\88\80G1,L1,T1. â¦\83G1, L1, T1â¦\84 â\8a\90[b] â¦\83G2, L2, T2â¦\84 → Q G1 L1 T1) →
- (â\88\80G1,G,L1,L,T1,T. â¦\83G1, L1, T1â¦\84 â\8a\90[b] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G, L, Tâ¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84 → Q G L T → Q G1 L1 T1) →
- â\88\80G1,L1,T1. â¦\83G1, L1, T1â¦\84 â\8a\90+[b] â¦\83G2, L2, T2â¦\84 → Q G1 L1 T1.
+ (â\88\80G1,L1,T1. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© → Q G1 L1 T1) →
+ (â\88\80G1,G,L1,L,T1,T. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G,L,Tâ\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© → Q G L T → Q G1 L1 T1) →
+ â\88\80G1,L1,T1. â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
qed-.
+(* Advanced properties ******************************************************)
+
+lemma fqup_zeta (b) (p) (I) (G) (K) (V):
+ ∀T1,T2. ⇧[1]T2 ≘ T1 → ❨G,K,ⓑ[p,I]V.T1❩ ⬂+[b] ❨G,K,T2❩.
+* /4 width=5 by fqup_strap2, fqu_fqup, fqu_drop, fqu_clear, fqu_bind_dx/ qed.
+
(* Basic_2A1: removed theorems 1: fqup_drop *)