notational change for lexs

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
index 14cce04dbf9624f66156f11ae1e8e5aab7df5137..a42aaa4654afd5ce7915da6bd1fe31f44c697ab9 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
@@ -18,37 +18,40 @@ include "basic_2/relocation/drops.ma".
 
 (* Properties with entrywise extension of context-sensitive relations *******)
 
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *)
-lemma lexs_deliftable_dropable: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
-                                dropable_sn (lexs RN RP).
-#RN #RP #HN #HP #b #f #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #X #f2 #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
+(* Basic_2A1: includes: lpx_sn_deliftable_dropable *) (**) (* changed after commit 13218 *)
+lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
+#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
   /4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #f #I #L1 #K1 #V1 #_ #IH #X #f2 #H #f1 #Hf2 elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ]
-  #g2 #Hg2 #H2 destruct elim (lexs_inv_next1 … H) -H
-  #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
-  /3 width=3 by drops_drop, ex2_intro/
-| #f #I #L1 #K1 #V1 #W1 #HLK #HWV #IH #X #f2 #H #f1 #Hf2 elim (after_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ]
-  #g1 #g2 #Hg2 #H1 #H2 destruct
-  [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H
-  #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
-  [ elim (HP … HV12 … HLK … HWV) | elim (HN … HV12 … HLK … HWV) ] -V1
-  /3 width=5 by lexs_next, lexs_push, drops_skip, ex2_intro/
+| #f #I #L1 #K1 #V1 #_ #IH #Hf #f2 #X #H #f1 #Hf2
+  elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct
+  elim (lexs_inv_push1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+  elim (IH … HL12 … Hg2) -g2
+  /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/
+| #f #I #L1 #K1 #V1 #W1 #HLK #HWV #IH #Hf #f2 #X #H #f1 #Hf2
+  lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+  lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct
+  lapply (lifts_fwd_isid … HWV … Hf) -HWV #H0 destruct
+  elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+  [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #L2 #V2 #HL12 #HV12 #H destruct 
+  elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2
+  lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct
+  /4 width=3 by drops_refl, lexs_next, lexs_push, isid_push, ex2_intro/
 ]
 qed-.
 
 (* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
-lemma lexs_liftable_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
-                                d_liftable2 RN → d_liftable2 RP → dedropable_sn (lexs RN RP).
+lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+                                      d_liftable2_sn RN → d_liftable2_sn RP → co_dedropable_sn (lexs RN RP).
 #RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
 [ #f #Hf #X #f1 #H #f2 #Hf2 >(lexs_inv_atom1 … H) -X
   /4 width=4 by drops_atom, lexs_atom, ex3_intro/
 | #f #I #L1 #K1 #V1 #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2
-  elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+  elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
   elim (IHLK1 … HK12 … Hg2) -K1
-  /3 width=6 by drops_drop, lexs_next, ex3_intro/
+  /3 width=6 by drops_drop, lexs_next, lexs_push, ex3_intro/
 | #f #I #L1 #K1 #V1 #W1 #HLK1 #HWV1 #IHLK1 #X #f1 #H #f2 #Hf2
-  elim (after_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+  elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
   [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #K2 #W2 #HK12 #HW12 #H destruct
   [ elim (H2RP … HW12 … HLK1 … HWV1) | elim (H2RN … HW12 … HLK1 … HWV1) ] -W1
   elim (IHLK1 … HK12 … Hg2) -K1
@@ -56,19 +59,19 @@ lemma lexs_liftable_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L.
 ]
 qed-.
 
-fact lexs_dropable_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
-                        ∀f2,L1. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f ⊚ f1 ≡ f2 →
-                        ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[RN, RP, f1] K2.
+fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
+                           ∀f2,L1. L1 ⪤*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≡ f2 →
+                           ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⪤*[RN, RP, f1] K2.
 #RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
   #H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/
 | #f #I #L2 #K2 #V2 #_ #IH #Hf #f2 #X #HX #f1 #Hf2
-  elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
-  elim (lexs_inv_next2 … HX) -HX #L1 #V1 #HL12 #HV12 #H destruct
+  elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+  elim (lexs_inv_push2 … HX) -HX #L1 #V1 #HL12 #HV12 #H destruct
   elim (IH … HL12 … Hg2) -L2 -V2 -g2
   /3 width=3 by drops_drop, isuni_inv_next, ex2_intro/
 | #f #I #L2 #K2 #V2 #W2 #_ #HWV2 #IH #Hf #f2 #X #HX #f1 #Hf2
-  elim (after_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+  elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
   [ elim (lexs_inv_push2 … HX) | elim (lexs_inv_next2 … HX) ] -HX #L1 #V1 #HL12 #HV12 #H destruct
   elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by isuni_fwd_push/ #K1 #HLK1 #HK12
   lapply (isuni_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf
@@ -79,73 +82,81 @@ fact lexs_dropable_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄
 qed-.
 
 (* Basic_2A1: includes: lpx_sn_dropable *)
-lemma lexs_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
-/2 width=5 by lexs_dropable_aux/ qed-.
+lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP).
+/2 width=5 by lexs_dropable_dx_aux/ qed-.
 
-(* Basic_2A1: includes: lpx_sn_drop_conf *)
-lemma lexs_drops_conf_next: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
-                            â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
-                            ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
-                            ∀f1. f ⊚ ⫯f1 ≡ f2 →
-                            â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
-#RN #RP #HRN #HRP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+(* Basic_2A1: includes: lpx_sn_drop_conf *) (**)
+lemma lexs_drops_conf_next: ∀RN,RP.
+                            â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+                            ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
+                            ∀f1. f ~⊚ ⫯f1 ≡ f2 →
+                            â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
 #X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX
 #K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma lexs_drops_conf_push: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
-                            â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
-                            ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
-                            ∀f1. f ⊚ ↑f1 ≡ f2 →
-                            â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
-#RN #RP #HRN #HRP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+lemma lexs_drops_conf_push: ∀RN,RP.
+                            â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+                            ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
+                            ∀f1. f ~⊚ ↑f1 ≡ f2 →
+                            â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
 #X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX
 #K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
 (* Basic_2A1: includes: lpx_sn_drop_trans *)
-lemma lexs_drops_trans_next: â\88\80RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+lemma lexs_drops_trans_next: â\88\80RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
                              ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
-                             ∀f1. f ⊚ ⫯f1 ≡ f2 →
-                             â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
+                             ∀f1. f ~⊚ ⫯f1 ≡ f2 →
+                             â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
 #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma lexs_drops_trans_push: â\88\80RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+lemma lexs_drops_trans_push: â\88\80RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
                              ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
-                             ∀f1. f ⊚ ↑f1 ≡ f2 →
-                             â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
+                             ∀f1. f ~⊚ ↑f1 ≡ f2 →
+                             â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
 #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
 lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
-                             d_liftable2 RN → d_liftable2 RP →
-                             â\88\80f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
+                             d_liftable2_sn RN → d_liftable2_sn RP →
+                             â\88\80f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
                              ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
-                             ∀f2. f ⊚ f1 ≡ ⫯f2 →
-                             â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⦻*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
+                             ∀f2. f ~⊚ f1 ≡ ⫯f2 →
+                             â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
 #X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX
 #L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
 qed-.
 
 lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
-                             d_liftable2 RN → d_liftable2 RP →
-                             â\88\80f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
+                             d_liftable2_sn RN → d_liftable2_sn RP →
+                             â\88\80f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
                              ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
-                             ∀f2. f ⊚ f1 ≡ ↑f2 →
-                             â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⦻*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
+                             ∀f2. f ~⊚ f1 ≡ ↑f2 →
+                             â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
 #X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX
 #L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
 qed-.
+
+lemma drops_atom2_lexs_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≡ ⋆ → 𝐔⦃f1⦄ →
+                             ∀f,L2. L1 ⪤*[RN, RP, f] L2 →
+                             ∀f2. f1 ~⊚ f2 ≡f → ⬇*[b, f1] L2 ≡ ⋆.
+#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
+elim (lexs_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
+#L #H #HL2 lapply (lexs_inv_atom1 … H) -H //
+qed-.