renaming

[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 2bdf99d9ec1c7a5fb2babe707135a3da25abf55d..11b08f3e665021fc6f4729c03e764a319f1f17ed 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
@@ -12,13 +12,14 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/relocation/lifts_lifts_bind.ma".
 include "basic_2/relocation/drops.ma".
 
 (* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
 
 (* Properties with entrywise extension of context-sensitive relations *******)
 
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *) (**) (* changed after commit 13218 *)
+(**) (* 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
@@ -40,9 +41,36 @@ lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
 ]
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
+lemma lexs_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+                                      ∀f2,L1,L2. L1 ⪤*[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
+                                      ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+                                      f ~⊚ f1 ≘ f2 → L1 ⪤*[RN, RP, f2] L2.
+#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
+#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
+[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
+  elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+  elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+  elim (lexs_inv_next … HK12) -HK12 #HK12 #HJ12
+  elim (HRN … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+  >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_next/
+| elim (coafter_inv_xxp … H) [1,2: |*: // ] -H *
+  [ #g #g1 #Hg2 #H1 #H2 destruct
+    elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+    elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+    elim (lexs_inv_push … HK12) -HK12 #HK12 #HJ12
+    elim (HRP … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+    >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_push/
+  | #g #Hg2 #H destruct
+    lapply (drops_inv_drop1 … HY1) -HY1 #HLK1
+    lapply (drops_inv_drop1 … HY2) -HY2 #HLK2
+    /3 width=9 by lexs_push/
+  ]
+]
+qed-.
+
 lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
-                                      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → co_dedropable_sn (lexs RN RP).
+                                      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb 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/
@@ -59,9 +87,9 @@ lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) →
 ]
 qed-.
 
-fact lexs_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â¬\87*[b, f] L2 â\89¡ K2 → 𝐔⦃f⦄ →
-                           â\88\80f2,L1. L1 ⪤*[RN, RP, f2] L2 â\86\92 â\88\80f1. f ~â\8a\9a f1 â\89¡ f2 →
-                           â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⪤*[RN, RP, f1] K2.
+fact lexs_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â¬\87*[b, f] L2 â\89\98 K2 → 𝐔⦃f⦄ →
+                           â\88\80f2,L1. L1 ⪤*[RN, RP, f2] L2 â\86\92 â\88\80f1. f ~â\8a\9a f1 â\89\98 f2 →
+                           â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89\98 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/
@@ -81,16 +109,14 @@ fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f
 ]
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_dropable *)
 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.
                             ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
-                            â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89¡ K1.ⓘ{I1} → 𝐔⦃f⦄ →
-                            â\88\80f1. f ~â\8a\9a ⫯f1 â\89¡ f2 →
-                            â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89¡ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+                            â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89\98 K1.ⓘ{I1} → 𝐔⦃f⦄ →
+                            â\88\80f1. f ~â\8a\9a â\86\91f1 â\89\98 f2 →
+                            â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89\98 K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #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
@@ -99,20 +125,19 @@ qed-.
 
 lemma lexs_drops_conf_push: ∀RN,RP.
                             ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
-                            â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89¡ K1.ⓘ{I1} → 𝐔⦃f⦄ →
-                            â\88\80f1. f ~â\8a\9a â\86\91f1 â\89¡ f2 →
-                            â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89¡ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+                            â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89\98 K1.ⓘ{I1} → 𝐔⦃f⦄ →
+                            â\88\80f1. f ~â\8a\9a ⫯f1 â\89\98 f2 →
+                            â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89\98 K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #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
 #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_drop_trans *)
 lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
-                             â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89¡ K2.ⓘ{I2} → 𝐔⦃f⦄ →
-                             â\88\80f1. f ~â\8a\9a ⫯f1 â\89¡ f2 →
-                             â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89¡ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+                             â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89\98 K2.ⓘ{I2} → 𝐔⦃f⦄ →
+                             â\88\80f1. f ~â\8a\9a â\86\91f1 â\89\98 f2 →
+                             â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89\98 K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
 elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
@@ -120,9 +145,9 @@ elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
 qed-.
 
 lemma lexs_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
-                             â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89¡ K2.ⓘ{I2} → 𝐔⦃f⦄ →
-                             â\88\80f1. f ~â\8a\9a â\86\91f1 â\89¡ f2 →
-                             â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89¡ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+                             â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89\98 K2.ⓘ{I2} → 𝐔⦃f⦄ →
+                             â\88\80f1. f ~â\8a\9a ⫯f1 â\89\98 f2 →
+                             â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89\98 K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
 elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
@@ -132,9 +157,9 @@ qed-.
 lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
                              d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
                              ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
-                             â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} â\89¡ K1 →
-                             â\88\80f2. f ~â\8a\9a f1 â\89¡ ⫯f2 →
-                             â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+                             â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} â\89\98 K1 →
+                             â\88\80f2. f ~â\8a\9a f1 â\89\98 â\86\91f2 →
+                             â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89\98 K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
 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
@@ -144,18 +169,18 @@ qed-.
 lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
                              d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
                              ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
-                             â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} â\89¡ K1 →
-                             â\88\80f2. f ~â\8a\9a f1 â\89¡ â\86\91f2 →
-                             â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+                             â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} â\89\98 K1 →
+                             â\88\80f2. f ~â\8a\9a f1 â\89\98 ⫯f2 →
+                             â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89\98 K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
 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
 #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
 qed-.
 
-lemma drops_atom2_lexs_conf: â\88\80RN,RP,b,f1,L1. â¬\87*[b, f1] L1 â\89¡ ⋆ → 𝐔⦃f1⦄ →
+lemma drops_atom2_lexs_conf: â\88\80RN,RP,b,f1,L1. â¬\87*[b, f1] L1 â\89\98 ⋆ → 𝐔⦃f1⦄ →
                              ∀f,L2. L1 ⪤*[RN, RP, f] L2 →
-                             â\88\80f2. f1 ~â\8a\9a f2 â\89¡f â\86\92 â¬\87*[b, f1] L2 â\89¡ ⋆.
+                             â\88\80f2. f1 ~â\8a\9a f2 â\89\98f â\86\92 â¬\87*[b, f1] L2 â\89\98 ⋆.
 #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 //