include "ground_2/relocation/rtmap_id.ma".
include "basic_2/notation/relations/relationstar_4.ma".
-include "basic_2/grammar/ceq.ma".
include "basic_2/relocation/lexs.ma".
include "basic_2/static/frees.ma".
interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-(* Basic properties ***********************************************************)
+definition R_frees_confluent: predicate (relation3 lenv term term) ≝
+ λRN.
+ ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
+ ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
+
+definition lexs_frees_confluent: relation (relation3 lenv term term) ≝
+ λRN,RP.
+ ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
+ ∀L2. 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 →
+ ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+(* Basic properties *********************************************************)
lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
#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.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
+ ∀V2. R L1 V V2 →
+ L1.ⓑ{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-.
+
+lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (lfxs R T).
+#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
+/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/
+qed-.
+
lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2,T. L1 ⦻*[R1, T] L2 → 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 → 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: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
+lemma lfxs_inv_atom_sn: ∀R,Y2,T. ⋆ ⦻*[R, T] Y2 → Y2 = ⋆.
+#R #Y2 #T * /2 width=4 by lexs_inv_atom1/
qed-.
-lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma lfxs_inv_atom_dx: ∀R,Y1,T. Y1 ⦻*[R, T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
qed-.
+lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,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/
+| lapply (frees_inv_sort … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
∃∃I,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 *
qed-.
lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
∃∃I,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 *
]
qed-.
-lemma lfxs_inv_bind: ∀R,I,L1,L2,V1,V2,T,p. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,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/
+| lapply (frees_inv_gref … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #T #p * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
+#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-.
(* Advanced inversion lemmas ************************************************)
+lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
+ ∃∃L2,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: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,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: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
Y2 = L2.ⓑ{I}V2.
]
qed-.
+lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
+ ∃∃L2,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: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,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/
+]
+qed-.
+
(* Basic forward lemmas *****************************************************)
-lemma lfxs_fwd_bind_sn: ∀R,I,L1,L2,V,T,p. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #I #L1 #L2 #V #T #p * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
qed-.
-lemma lfxs_fwd_bind_dx: ∀R,I,L1,L2,V1,V2,T,p. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
+lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #T #p #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
+#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
qed-.
lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
qed-.
+lemma lfxs_fwd_dx: ∀R,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
+/2 width=3 by ex1_2_intro/
+qed-.
+
(* Basic_2A1: removed theorems 24:
llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
llpx_sn_bind llpx_sn_flat
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