include "ground_2/relocation/rtmap_id.ma".
include "basic_2/notation/relations/relationstar_4.ma".
-include "basic_2/syntax/lenv_ext2.ma".
+include "basic_2/syntax/cext2.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).
-definition R_frees_confluent: predicate (relation3 …) ≝
- 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 …) ≝
- 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.
∀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}] ⋆.
-#R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/
-qed.
-
-(* Basic_2A1: uses: llpx_sn_sort *)
-lemma lfxs_sort: ∀R,I1,I2,L1,L2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓘ{I1} ⪤*[R, ⋆s] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
-lapply (frees_inv_sort … Hf) -Hf
-/4 width=3 by frees_sort, lexs_push, isid_push, ex2_intro/
-qed.
-
-lemma lfxs_pair: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
-#R #I1 #I2 #L1 #L2 #V1 *
-/4 width=3 by ext2_pair, frees_pair, lexs_next, ex2_intro/
-qed.
-
-lemma lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
-/4 width=3 by frees_unit, lexs_next, ext2_unit, ex2_intro/ qed.
-
-lemma lfxs_lref: ∀R,I1,I2,L1,L2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #⫯i] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
-qed.
-
-(* Basic_2A1: uses: llpx_sn_gref *)
-lemma lfxs_gref: ∀R,I1,I2,L1,L2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓘ{I1} ⪤*[R, §l] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
-lapply (frees_inv_gref … Hf) -Hf
-/4 width=3 by frees_gref, lexs_push, isid_push, ex2_intro/
-qed.
-
-lemma lfxs_bind_repl_dx: ∀R,I,I1,L1,L2,T.
- L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I1} →
- ∀I2. cext2 R L1 I I2 →
- L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I2}.
-#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
-/3 width=5 by lexs_pair_repl, ex2_intro/
-qed-.
-
-lemma lfxs_sym: ∀R. lexs_frees_confluent (cext2 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
-/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
-qed-.
+definition lfxs_confluent: relation … ≝
+ λR1,R2.
+ ∀K1,K,V1. K1 ⪤*[R1, V1] K → ∀V. R1 K1 V1 V →
+ ∀K2. K ⪤*[R2, V] K2 → K ⪤*[R2, V1] K2.
-(* Basic_2A1: uses: llpx_sn_co *)
-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 * /5 width=7 by lexs_co, cext2_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-.
+definition lfxs_transitive: relation3 ? (relation3 ?? term) … ≝
+ λR1,R2,R3.
+ ∀K1,K,V1. K1 ⪤*[R1, V1] K →
+ ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
(* Basic inversion lemmas ***************************************************)
(* Basic forward lemmas *****************************************************)
+lemma lfxs_fwd_zero_pair: ∀R,I,K1,K2,V1,V2.
+ K1.ⓑ{I}V1 ⪤*[R, #0] K2.ⓑ{I}V2 → K1 ⪤*[R, V1] K2.
+#R #I #K1 #K2 #V1 #V2 #H
+elim (lfxs_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
+qed-.
+
(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
/2 width=3 by ex1_2_intro/
qed-.
+(* Basic properties *********************************************************)
+
+lemma lfxs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
+#R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/
+qed.
+
+(* Basic_2A1: uses: llpx_sn_sort *)
+lemma lfxs_sort: ∀R,I1,I2,L1,L2,s.
+ L1 ⪤*[R, ⋆s] L2 → L1.ⓘ{I1} ⪤*[R, ⋆s] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
+lapply (frees_inv_sort … Hf) -Hf
+/4 width=3 by frees_sort, lexs_push, isid_push, ex2_intro/
+qed.
+
+lemma lfxs_pair: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
+#R #I1 #I2 #L1 #L2 #V1 *
+/4 width=3 by ext2_pair, frees_pair, lexs_next, ex2_intro/
+qed.
+
+lemma lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
+ L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+/4 width=3 by frees_unit, lexs_next, ext2_unit, ex2_intro/ qed.
+
+lemma lfxs_lref: ∀R,I1,I2,L1,L2,i.
+ L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #⫯i] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
+qed.
+
+(* Basic_2A1: uses: llpx_sn_gref *)
+lemma lfxs_gref: ∀R,I1,I2,L1,L2,l.
+ L1 ⪤*[R, §l] L2 → L1.ⓘ{I1} ⪤*[R, §l] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
+lapply (frees_inv_gref … Hf) -Hf
+/4 width=3 by frees_gref, lexs_push, isid_push, ex2_intro/
+qed.
+
+lemma lfxs_bind_repl_dx: ∀R,I,I1,L1,L2,T.
+ L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I1} →
+ ∀I2. cext2 R L1 I I2 →
+ L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I2}.
+#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #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) →
+ ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by lexs_co, cext2_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-.
+
+lemma lfxs_unit_sn: ∀R1,R2,I,K1,L2.
+ K1.ⓤ{I} ⪤*[R1, #0] L2 → K1.ⓤ{I} ⪤*[R2, #0] L2.
+#R1 #R2 #I #K1 #L2 #H
+elim (lfxs_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
+/3 width=7 by lfxs_unit, lexs_co_isid/
+qed-.
+
(* Basic_2A1: removed theorems 9:
llpx_sn_skip llpx_sn_lref llpx_sn_free
llpx_sn_fwd_lref
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