X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flexs.ma;h=0ff659b1376689e45afb7fdca00469607a70a691;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=b002e073af1af7be489b4f845d88376be49088cf;hpb=632c54beaf67e68a1eeeec22274466157003b779;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
index b002e073a..0ff659b13 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
@@ -13,245 +13,284 @@
 (**************************************************************************)
 
 include "ground_2/relocation/rtmap_sle.ma".
+include "ground_2/relocation/rtmap_sdj.ma".
 include "basic_2/notation/relations/relationstar_5.ma".
 include "basic_2/syntax/lenv.ma".
 
 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
 
-(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
-inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝
+inductive lexs (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
 | lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
-| lexs_next: ∀f,I,L1,L2,V1,V2.
-             lexs RN RP f L1 L2 → RN L1 V1 V2 →
-             lexs RN RP (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
-| lexs_push: ∀f,I,L1,L2,V1,V2.
-             lexs RN RP f L1 L2 → RP L1 V1 V2 →
-             lexs RN RP (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+| 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_push: ∀f,I1,I2,L1,L2.
+             lexs RN RP f L1 L2 → RP L1 I1 I2 →
+             lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
 .
 
 interpretation "generic entrywise extension (local environment)"
    'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2).
 
-definition R_pw_confluent2_lexs: relation3 lenv term term → relation3 lenv term term →
-                                 relation3 lenv term term → relation3 lenv term term →
-                                 relation3 lenv term term → relation3 lenv term term →
-                                 relation3 rtmap lenv term ≝
-                                 λR1,R2,RN1,RP1,RN2,RP2,f,L0,T0.
-                                 ∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
-                                 ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
-                                 ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lexs_transitive: relation5 (relation3 lenv term term)
-                                      (relation3 lenv term term) … ≝
-                            λR1,R2,R3,RN,RP.
-                            ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⦻*[RN, RP, f] L2 →
-                            ∀T2. R2 L2 T T2 → R3 L1 T1 T2.
+definition R_pw_confluent2_lexs: relation3 lenv bind bind → relation3 lenv bind bind →
+                                 relation3 lenv bind bind → relation3 lenv bind bind →
+                                 relation3 lenv bind bind → relation3 lenv bind bind →
+                                 relation3 rtmap lenv bind ≝
+                                 λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
+                                 ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
+                                 ∀L1. L0 ⪤*[RN1, RP1, f] L1 → ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
+                                 ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
+
+definition lexs_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
+                            relation3 lenv bind bind →
+                            relation3 lenv bind bind → relation3 lenv bind bind →
+                            relation3 rtmap lenv bind ≝
+                            λR1,R2,R3,RN,RP,f,L1,I1.
+                            ∀I. R1 L1 I1 I → ∀L2. L1 ⪤*[RN, RP, f] L2 →
+                            ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
-#f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⦻*[RN, RP, f] Y → Y = ⋆.
+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: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g →
-                         ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
+                         ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
 #RN #RP #f #X #Y * -f -X -Y
-[ #f #g #J #K1 #W1 #H destruct
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_next … H2) -g destruct
+[ #f #g #J1 #K1 #H destruct
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
   /2 width=5 by ex3_2_intro/
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H)
 ]
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y →
-                      ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ↑g] 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: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g →
-                         ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → 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 #J #K1 #W1 #H destruct
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_next_push … H)
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_push … H2) -g destruct
+[ #f #g #J1 #K1 #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
   /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma lexs_inv_push1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y →
-                      ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{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-.
 
-fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
-#f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⦻*[RN, RP, f] ⋆ → X = ⋆.
+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: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g →
-                         ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
+                         ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
 #RN #RP #f #X #Y * -f -X -Y
-[ #f #g #J #K2 #W2 #H destruct
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_next … H2) -g destruct
+[ #f #g #J2 #K2 #H destruct
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
   /2 width=5 by ex3_2_intro/
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H)
 ]
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
-                      ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ↑g] 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: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g →
-                         ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → 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 #J #K2 #W2 #g #H destruct
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_next_push … H)
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_push … H2) -g destruct
+[ #f #J2 #K2 #g #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
   /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma lexs_inv_push2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 →
-                      ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_push2: ∀RN,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,V1,V2.
-                     L1.ⓑ{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
-                     ∧∧ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
-#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_next1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
+                     L1.ⓘ{I1} ⪤*[RN, RP, ↑f] 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,V1,V2.
-                     L1.ⓑ{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
-                     ∧∧ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
-#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
+                     L1.ⓘ{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/
 qed-.
 
-lemma lexs_inv_tl: ∀RN,RP,f,I,L1,L2,V1,V2. L1 ⦻*[RN, RP, ⫱f] L2 →
-                   RN L1 V1 V2 → RP L1 V1 V2 → 
-                   L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2.
-#RN #RP #f #I #L2 #L2 #V1 #V2 elim (pn_split f) *
+lemma lexs_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤*[RN, RP, ⫱f] L2 →
+                   RN L1 I1 I2 → RP L1 I1 I2 → 
+                   L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2}.
+#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
 /2 width=1 by lexs_next, lexs_push/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lexs_fwd_pair: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. 
-                     L1.ⓑ{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 →
-                     L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2.
-#RN #RP #f #I1 #I2 #L2 #L2 #V1 #V2 #Hf
+lemma lexs_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2. 
+                     L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2} →
+                     L1 ⪤*[RN, RP, ⫱f] L2.
+#RN #RP #f #I1 #I2 #L1 #L2 #Hf
 elim (pn_split f) * #g #H destruct
-[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf
-/2 width=1 by conj/
+[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤*[RN, RP, f] L2).
 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H
+#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
 [ elim (eq_inv_nx … H) -H /3 width=3 by lexs_next/
 | elim (eq_inv_px … H) -H /3 width=3 by lexs_push/
 ]
 qed-.
 
-lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤*[RN, RP, f] L2).
 #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_refl *)
-lemma lexs_refl: ∀RN,RP.
-                 (∀L. reflexive … (RN L)) →
-                 (∀L. reflexive … (RP L)) →
+lemma lexs_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
                  ∀f.reflexive … (lexs RN RP f).
 #RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
-#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/
+#L #I #IH #f elim (pn_split f) *
+#g #H destruct /2 width=1 by lexs_next, lexs_push/
 qed.
 
 lemma lexs_sym: ∀RN,RP.
-                (∀L1,L2,T1,T2. RN L1 T1 T2 → RN L2 T2 T1) →
-                (∀L1,L2,T1,T2. RP L1 T1 T2 → RP L2 T2 T1) →
+                (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
+                (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
                 ∀f. symmetric … (lexs RN RP f).
 #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
 /3 width=2 by lexs_next, lexs_push/
 qed-.
 
-lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2.
-                      L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 →
-                      ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 →
-                      L1.ⓑ{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2.
-#RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP
-elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/
-qed-.
+lemma lexs_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
+                      L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2} →
+                      ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
+                      L1.ⓘ{J1} ⪤*[RN, RP, f] L2.ⓘ{J2}.
+/3 width=3 by lexs_inv_tl, lexs_fwd_bind/ qed-.
 
-lemma lexs_co: ∀RN1,RP1,RN2,RP2.
-               (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
-               (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
-               ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2.
+lemma lexs_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
+               ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → L1 ⪤*[RN2, RP2, f] L2.
 #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
 /3 width=1 by lexs_atom, lexs_next, lexs_push/
 qed-.
 
-lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
-                    (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
-                    ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
-                    L1 ⦻*[RN2, RP2, f] L2.
+lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
+                    ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
+                    L1 ⪤*[RN2, RP2, f] L2.
 #RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
-#f #I #K1 #K2 #V1 #V2 #_ #HV12 #IH #H
+#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
 [ elim (isid_inv_next … H) -H //
 | /4 width=3 by lexs_push, isid_inv_push/
-] 
+]
 qed-.
 
-lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
-                      ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
-                      ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2.
+lemma lexs_sdj: ∀RN,RP. RP ⊆ RN →
+                ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+                ∀f2. f1 ∥ f2 → L1 ⪤*[RP, RN, f2] L2.
+#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
+[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
+  #g2 #H #H2 destruct /3 width=1 by lexs_push/
+| elim (sdj_inv_px … H12) -H12 [2,4: // ] *
+  #g2 #H #H2 destruct /3 width=1 by lexs_next, lexs_push/
+]
+qed-.
+
+lemma sle_lexs_trans: ∀RN,RP. RN ⊆ RP →
+                      ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+                      ∀f1. f1 ⊆ f2 → L1 ⪤*[RN, RP, f1] L2.
 #RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
-#f2 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
-[ * * [2: #n1 ] ] #f1 #H
-[ /4 width=5 by lexs_next, sle_inv_nn/
-| /4 width=5 by lexs_push, sle_inv_pn/
-| elim (sle_inv_xp … H) -H [2,3: // ]
+#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
+[ elim (pn_split f1) * ]
+[ /4 width=5 by lexs_push, sle_inv_pn/
+| /4 width=5 by lexs_next, sle_inv_nn/
+| elim (sle_inv_xp … H12) -H12 [2,3: // ]
   #g1 #H #H1 destruct /3 width=5 by lexs_push/
 ]
 qed-.
 
-lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
-                     ∀f1,L1,L2. L1 ⦻*[RN, RP, f1] L2 →
-                     ∀f2. f1 ⊆ f2 → L1 ⦻*[RN, RP, f2] L2.
+lemma sle_lexs_conf: ∀RN,RP. RP ⊆ RN →
+                     ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+                     ∀f2. f1 ⊆ f2 → L1 ⪤*[RN, RP, f2] L2.
 #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
-#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
-[2: * * [2: #n2 ] ] #f2 #H
-[ /4 width=5 by lexs_next, sle_inv_pn/
-| /4 width=5 by lexs_push, sle_inv_pp/
-| elim (sle_inv_nx … H) -H [2,3: // ]
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
+[2: elim (pn_split f2) * ]
+[ /4 width=5 by lexs_push, sle_inv_pp/
+| /4 width=5 by lexs_next, sle_inv_pn/
+| elim (sle_inv_nx … H12) -H12 [2,3: // ]
   #g2 #H #H2 destruct /3 width=5 by lexs_next/
 ]
 qed-.
 
-lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                      ∀f,L1,L2. L1 ⦻*[R1, RP, f] L2 → ∀g. f ⊆ g →
-                      ∃∃L. L1 ⦻*[R1, RP, g] L & L ⦻*[R2, cfull, f] L2.
+lemma lexs_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+                      ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g →
+                      ∃∃L. L1 ⪤*[R1, RP, g] L & L ⪤*[R2, cfull, f] L2.
 #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
 [ /2 width=3 by lexs_atom, ex2_intro/ ]
-#f #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #y #H
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
 [ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, ex2_intro/
 | elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
 ]
 qed-.
+
+lemma lexs_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+                      ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ∥ g →
+                      ∃∃L. L1 ⪤*[RP, R1, g] L & L ⪤*[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by lexs_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+  elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+  elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+]
+qed-.
+
+lemma lexs_dec: ∀RN,RP.
+                (∀L,I1,I2. Decidable (RN L I1 I2)) →
+                (∀L,I1,I2. Decidable (RP L I1 I2)) →
+                ∀L1,L2,f. Decidable (L1 ⪤*[RN, RP, f] L2).
+#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
+[ /2 width=1 by lexs_atom, or_introl/
+| #L2 #I2 #f @or_intror #H
+  lapply (lexs_inv_atom1 … H) -H #H destruct
+| #f @or_intror #H
+  lapply (lexs_inv_atom2 … H) -H #H destruct
+| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12
+  [2: /4 width=3 by lexs_fwd_bind, or_intror/ ]
+  elim (pn_split f) * #g #H destruct
+  [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
+  [1,3: /3 width=1 by lexs_push, lexs_next, or_introl/ ]
+  @or_intror #H
+  [ elim (lexs_inv_push … H) | elim (lexs_inv_next … H) ] -H
+  /2 width=1 by/
+]
+qed-.