include "ground_2/relocation/rtmap_sand.ma".
include "ground_2/relocation/rtmap_sor.ma".
-include "basic_2/grammar/lenv_weight.ma".
include "basic_2/relocation/lexs.ma".
+include "basic_2/relocation/drops.ma".
(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
(* Main properties **********************************************************)
-(* Basic_2A1: includes: lpx_sn_trans *)
-theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RP →
- lexs_transitive RP RP RN RP →
- Transitive … (lexs RN RP f).
-#RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
+theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
+ lexs_transitive RN1 RN2 RN RN1 RP1 →
+ lexs_transitive RP1 RP2 RP RN1 RP1 →
+ ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
+ ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
+ L1 ⦻*[RN, RP, f] L2.
+#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
-| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
+| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
#K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
-| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
+| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
#K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
]
qed-.
+(* Basic_2A1: includes: lpx_sn_trans *)
+theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP →
+ lexs_transitive RP RP RP RN RP →
+ Transitive … (lexs RN RP f).
+/2 width=9 by lexs_trans_gen/ qed-.
+
(* Basic_2A1: includes: lpx_sn_conf *)
-theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
- lpx_sn_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
- lpx_sn_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
- ∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f).
-#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 generalize in match f; -f
-@(f_ind … lw … L0) -L0 #x #IH *
-[ #_ #f #X1 #H1 #X2 #H2 -x
- >(lexs_inv_atom1 … H1) -X1
- >(lexs_inv_atom1 … H2) -X2 /2 width=3 by lexs_atom, ex2_intro/
-| #L0 #I #V0 #Hx #f elim (pn_split f) *
- #g #H #X1 #H1 #X2 #H2 destruct
- [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
- elim (HRP … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_push, ex2_intro/
- | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
- elim (HRN … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_next, ex2_intro/
+theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
+ ∀L,f.
+ (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ⫯g = ⫱*[n] f → lexs_pw_confluent2_R RN1 RN2 RN1 RP1 RN2 RP2 g K V) →
+ (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ↑g = ⫱*[n] f → lexs_pw_confluent2_R RP1 RP2 RN1 RP1 RN2 RP2 g K V) →
+ pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
+#RN1 #RP1 #RN2 #RP2 #L elim L -L
+[ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
+ /2 width=3 by lexs_atom, ex2_intro/
+| #L #I #V #IH #f elim (pn_split f) * #g #H destruct
+ #HN #HP #Y1 #H1 #Y2 #H2
+ [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
+ elim (HP … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
+ | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
+ elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
+ elim (HN … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
]
]
qed-.
right_cancellable … (lexs RN RP f).
/3 width=3 by/ qed-.
-theorem lexs_meet: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
- ∀f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
-#g2 #H #H2 #f #Hf destruct
-[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
-#H2 #H2V #_
-[ elim (sand_inv_npx … Hf) | elim (sand_inv_ppx … Hf) | elim (sand_inv_nnx … Hf) | elim (sand_inv_pnx … Hf) ] -Hf
-/3 width=5 by lexs_next, lexs_push/
+lemma lexs_meet: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⦻*[RN, RP, f1] L2 →
+ ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
+[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
+| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
+] -Hf /3 width=5 by lexs_next, lexs_push/
qed-.
-theorem lexs_join: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
- ∀f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
-#g2 #H #H2 #f #Hf destruct
-[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
-#H2 #H2V #_
-[ elim (sor_inv_npx … Hf) | elim (sor_inv_ppx … Hf) | elim (sor_inv_nnx … Hf) | elim (sor_inv_pnx … Hf) ] -Hf
-/3 width=5 by lexs_next, lexs_push/
+lemma lexs_join: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⦻*[RN, RP, f1] L2 →
+ ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
+[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
+| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+] -Hf /3 width=5 by lexs_next, lexs_push/
qed-.
projects / helm.git / blobdiff