| lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
| lexs_next: ∀f,I1,I2,L1,L2.
lexs RN RP f L1 L2 → RN L1 I1 I2 →
- lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ lexs RN RP (â\86\91f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
| lexs_push: ∀f,I1,I2,L1,L2.
lexs RN RP f L1 L2 → RP L1 I1 I2 →
- lexs RN RP (â\86\91f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
.
interpretation "generic entrywise extension (local environment)"
lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤*[RN, RP, f] Y → Y = ⋆.
/2 width=6 by lexs_inv_atom1_aux/ qed-.
-fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = ⫯g →
+fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = â\86\91g →
∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, ⫯g] Y →
+lemma lexs_inv_next1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, â\86\91g] Y →
∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by lexs_inv_next1_aux/ qed-.
-fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = â\86\91g →
+fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = ⫯g →
∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
]
qed-.
-lemma lexs_inv_push1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, â\86\91g] Y →
+lemma lexs_inv_push1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, ⫯g] Y →
∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by lexs_inv_push1_aux/ qed-.
lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆.
/2 width=6 by lexs_inv_atom2_aux/ qed-.
-fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = ⫯g →
+fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = â\86\91g →
∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J2 #K2 #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} →
+lemma lexs_inv_next2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, â\86\91g] K2.ⓘ{J2} →
∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by lexs_inv_next2_aux/ qed-.
-fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = â\86\91g →
+fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = ⫯g →
∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #J2 #K2 #g #H destruct
]
qed-.
-lemma lexs_inv_push2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, â\86\91g] K2.ⓘ{J2} →
+lemma lexs_inv_push2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} →
∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by lexs_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
- L1.â\93\98{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} →
+ L1.â\93\98{I1} ⪤*[RN, RP, â\86\91f] L2.ⓘ{I2} →
L1 ⪤*[RN, RP, f] L2 ∧ RN L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_next1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
- L1.â\93\98{I1} ⪤*[RN, RP, â\86\91f] L2.ⓘ{I2} →
+ L1.â\93\98{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} →
L1 ⪤*[RN, RP, f] L2 ∧ RP L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_push1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/