(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition lfxs (R) (T): relation lenv ≝
- λL1,L2. â\88\83â\88\83f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f & L1 ⦻*[R, cfull, f] L2.
+ λL1,L2. â\88\83â\88\83f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f & L1 ⪤*[R, cfull, f] L2.
interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
definition lexs_frees_confluent: relation (relation3 lenv term term) ≝
λRN,RP.
∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
- â\88\80L2. L1 ⦻*[RN, RP, f1] L2 →
+ â\88\80L2. L1 ⪤*[RN, RP, f1] L2 →
∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
(relation3 lenv term term) … ≝
λR1,R2,RP1,RP2.
∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- â\88\80L1. L0 ⦻*[RP1, T0] L1 â\86\92 â\88\80L2. L0 ⦻*[RP2, T0] L2 →
+ â\88\80L1. L0 ⪤*[RP1, T0] L1 â\86\92 â\88\80L2. L0 ⪤*[RP2, T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
(* Basic properties *********************************************************)
-lemma lfxs_atom: â\88\80R,I. â\8b\86 ⦻*[R, ⓪{I}] ⋆.
+lemma lfxs_atom: â\88\80R,I. â\8b\86 ⪤*[R, ⓪{I}] ⋆.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
qed.
(* Basic_2A1: uses: llpx_sn_sort *)
lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⦻*[R, â\8b\86s] L2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2.
+ L1 ⪤*[R, â\8b\86s] L2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
qed.
-lemma lfxs_zero: â\88\80R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
- R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
+lemma lfxs_zero: â\88\80R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
+ R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
qed.
lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⦻*[R, #i] L2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2.
+ L1 ⪤*[R, #i] L2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, #⫯i] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
qed.
(* Basic_2A1: uses: llpx_sn_gref *)
lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⦻*[R, §l] L2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2.
+ L1 ⪤*[R, §l] L2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
qed.
lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.â\93\91{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
+ L1.â\93\91{I}V ⪤*[R, T] L2.ⓑ{I}V1 →
∀V2. R L1 V V2 →
- L1.â\93\91{I}V ⦻*[R, T] L2.ⓑ{I}V2.
+ L1.â\93\91{I}V ⪤*[R, T] L2.ⓑ{I}V2.
#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
/3 width=5 by lexs_pair_repl, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_co *)
lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- â\88\80L1,L2,T. L1 ⦻*[R1, T] L2 â\86\92 L1 ⦻*[R2, T] L2.
+ â\88\80L1,L2,T. L1 ⪤*[R1, T] L2 â\86\92 L1 ⪤*[R2, T] L2.
#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
qed-.
lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
(∀f. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → 𝐈⦃f⦄) →
(∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≡ f) →
- L1 ⦻*[R1, T1] L2 â\86\92 L1 ⦻*[R2, T2] L2.
+ L1 ⪤*[R1, T1] L2 â\86\92 L1 ⪤*[R2, T2] L2.
#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
/4 width=7 by lexs_co_isid, ex2_intro/
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma lfxs_inv_atom_sn: â\88\80R,Y2,T. â\8b\86 ⦻*[R, T] Y2 → Y2 = ⋆.
+lemma lfxs_inv_atom_sn: â\88\80R,Y2,T. â\8b\86 ⪤*[R, T] Y2 → Y2 = ⋆.
#R #Y2 #T * /2 width=4 by lexs_inv_atom1/
qed-.
-lemma lfxs_inv_atom_dx: â\88\80R,Y1,T. Y1 ⦻*[R, T] ⋆ → Y1 = ⋆.
+lemma lfxs_inv_atom_dx: â\88\80R,Y1,T. Y1 ⪤*[R, T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
qed-.
-lemma lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
+lemma lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤*[R, ⋆s] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma lfxs_inv_zero: â\88\80R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
+lemma lfxs_inv_zero: â\88\80R,Y1,Y2. Y1 ⪤*[R, #0] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma lfxs_inv_lref: â\88\80R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+lemma lfxs_inv_lref: â\88\80R,Y1,Y2,i. Y1 ⪤*[R, #⫯i] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤*[R, #i] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+lemma lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤*[R, §l] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
-lemma lfxs_inv_bind: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- L1 ⦻*[R, V1] L2 â\88§ L1.â\93\91{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+lemma lfxs_inv_bind: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ L1 ⪤*[R, V1] L2 â\88§ L1.â\93\91{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_flat *)
-lemma lfxs_inv_flat: â\88\80R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
- L1 ⦻*[R, V] L2 â\88§ L1 ⦻*[R, T] L2.
+lemma lfxs_inv_flat: â\88\80R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
+ L1 ⪤*[R, V] L2 â\88§ L1 ⪤*[R, T] L2.
#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfxs_inv_sort_pair_sn: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⦻*[R, ⋆s] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_sort_pair_sn: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⪤*[R, ⋆s] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
[ #H destruct
| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⪤*[R, ⋆s] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
[ #_ #H destruct
| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_zero_pair_sn: â\88\80R,I,Y2,L1,V1. L1.â\93\91{I}V1 ⦻*[R, #0] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+lemma lfxs_inv_zero_pair_sn: â\88\80R,I,Y2,L1,V1. L1.â\93\91{I}V1 ⪤*[R, #0] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
[ #H destruct
]
qed-.
-lemma lfxs_inv_zero_pair_dx: â\88\80R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+lemma lfxs_inv_zero_pair_dx: â\88\80R,I,Y1,L2,V2. Y1 ⪤*[R, #0] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
[ #_ #H destruct
]
qed-.
-lemma lfxs_inv_lref_pair_sn: â\88\80R,I,Y2,L1,V1,i. L1.â\93\91{I}V1 ⦻*[R, #⫯i] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_lref_pair_sn: â\88\80R,I,Y2,L1,V1,i. L1.â\93\91{I}V1 ⪤*[R, #⫯i] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
[ #H destruct
| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_lref_pair_dx: â\88\80R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_lref_pair_dx: â\88\80R,I,Y1,L2,V2,i. Y1 ⪤*[R, #⫯i] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
[ #_ #H destruct
| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⦻*[R, §l] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⪤*[R, §l] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
[ #H destruct
| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⪤*[R, §l] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
[ #_ #H destruct
| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
-lemma lfxs_fwd_pair_sn: â\88\80R,I,L1,L2,V,T. L1 ⦻*[R, â\91¡{I}V.T] L2 â\86\92 L1 ⦻*[R, V] L2.
+lemma lfxs_fwd_pair_sn: â\88\80R,I,L1,L2,V,T. L1 ⪤*[R, â\91¡{I}V.T] L2 â\86\92 L1 ⪤*[R, V] L2.
#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
-lemma lfxs_fwd_bind_dx: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+lemma lfxs_fwd_bind_dx: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
-lemma lfxs_fwd_flat_dx: â\88\80R,I,L1,L2,V,T. L1 ⦻*[R, â\93\95{I}V.T] L2 â\86\92 L1 ⦻*[R, T] L2.
+lemma lfxs_fwd_flat_dx: â\88\80R,I,L1,L2,V,T. L1 ⪤*[R, â\93\95{I}V.T] L2 â\86\92 L1 ⪤*[R, T] L2.
#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
qed-.
-lemma lfxs_fwd_dx: â\88\80R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 →
+lemma lfxs_fwd_dx: â\88\80R,I,L1,K2,T,V2. L1 ⪤*[R, T] K2.ⓑ{I}V2 →
∃∃K1,V1. L1 = K1.ⓑ{I}V1.
#R #I #L1 #K2 #T #V2 * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
[ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #K1 #V1 #_ #_ #H destruct
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