renaming

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lexs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
index acfd31ed520bda019b54eb92a116a67d270f00c6..0ff659b1376689e45afb7fdca00469607a70a691 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
@@ -13,20 +13,20 @@
 (**************************************************************************)
 
 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 bind bind): rtmap → relation lenv ≝
 | lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
 | 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 RN RP (â\86\91f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
 | lexs_push: ∀f,I1,I2,L1,L2.
              lexs RN RP f L1 L2 → RP L1 I1 I2 →
-             lexs RN RP (â\86\91f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+             lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
 .
 
 interpretation "generic entrywise extension (local environment)"
@@ -36,16 +36,18 @@ definition R_pw_confluent2_lexs: relation3 lenv bind bind → relation3 lenv bin
                                  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,T0.
-                                 ∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+                                 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 →
-                                 ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+                                 ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
 
-definition lexs_transitive: relation5 (relation3 lenv bind bind)
-                                      (relation3 lenv bind bind) … ≝
-                            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 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 ***************************************************)
 
@@ -58,7 +60,7 @@ qed-.
 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: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = ⫯g →
+fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = â\86\91g →
                          ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J1 #K1 #H destruct
@@ -69,11 +71,11 @@ fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X =
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, ⫯g] Y →
+lemma lexs_inv_next1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, â\86\91g] 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: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 f = â\86\91g →
+fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J1,K1. X = K1.â\93\98{J1} â\86\92 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 #J1 #K1 #H destruct
@@ -83,7 +85,7 @@ fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X =
 ]
 qed-.
 
-lemma lexs_inv_push1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{J1} ⪤*[RN, RP, â\86\91g] Y →
+lemma lexs_inv_push1: â\88\80RN,RP,g,J1,K1,Y. K1.â\93\98{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-.
 
@@ -96,7 +98,7 @@ qed-.
 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: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = ⫯g →
+fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = â\86\91g →
                          ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J2 #K2 #H destruct
@@ -107,11 +109,11 @@ fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y =
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} →
+lemma lexs_inv_next2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, â\86\91g] 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: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 f = â\86\91g →
+fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y â\86\92 â\88\80g,J2,K2. Y = K2.â\93\98{J2} â\86\92 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 #J2 #K2 #g #H destruct
@@ -121,20 +123,20 @@ fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y =
 ]
 qed-.
 
-lemma lexs_inv_push2: â\88\80RN,RP,g,J2,X,K2. X ⪤*[RN, RP, â\86\91g] K2.ⓘ{J2} →
+lemma lexs_inv_push2: â\88\80RN,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.
-                     L1.â\93\98{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} →
+                     L1.â\93\98{I1} ⪤*[RN, RP, â\86\91f] 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.
-                     L1.â\93\98{I1} ⪤*[RN, RP, â\86\91f] L2.ⓘ{I2} →
+                     L1.â\93\98{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/
@@ -171,10 +173,7 @@ lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤*[RN, RP, f]
 #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
 qed-.
 
-(* Basic_2A1: uses: 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 #IH #f elim (pn_split f) *
@@ -195,16 +194,13 @@ lemma lexs_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
                       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,I1,I2. RN1 L1 I1 I2 → RN2 L1 I1 I2) →
-               (∀L1,I1,I2. RP1 L1 I1 I2 → RP2 L1 I1 I2) →
+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,I1,I2. RP1 L1 I1 I2 → RP2 L1 I1 I2) →
+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 //
@@ -214,7 +210,19 @@ lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
 ]
 qed-.
 
-lemma sle_lexs_trans: ∀RN,RP. (∀L,I1,I2. RN L I1 I2 → RP L I1 I2) →
+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 //
@@ -227,7 +235,7 @@ lemma sle_lexs_trans: ∀RN,RP. (∀L,I1,I2. RN L I1 I2 → RP L I1 I2) →
 ]
 qed-.
 
-lemma sle_lexs_conf: ∀RN,RP. (∀L,I1,I2. RP L I1 I2 → RN L I1 I2) →
+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 //
@@ -240,7 +248,7 @@ lemma sle_lexs_conf: ∀RN,RP. (∀L,I1,I2. RP L I1 I2 → RN L I1 I2) →
 ]
 qed-.
 
-lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+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
@@ -253,6 +261,19 @@ lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. refle
 ]
 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)) →