X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fdrops_lexs.ma;h=11b08f3e665021fc6f4729c03e764a319f1f17ed;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=7cfbf82c03a0827e94ea132587c8fa700e3b99ef;hpb=c879284b576409cec07e96c1f08510d9d9ac14f3;p=helm.git

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 7cfbf82c0..11b08f3e6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
@@ -19,7 +19,7 @@ include "basic_2/relocation/drops.ma".
 
 (* 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
@@ -43,8 +43,8 @@ qed-.
 
 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.
+                                      ∀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
@@ -68,7 +68,6 @@ lemma lexs_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN →
 ]
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
 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).
@@ -88,9 +87,9 @@ lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) →
 ]
 qed-.
 
-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.
+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/
@@ -110,16 +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 →
-                            ∀b,f,I1,K1. ⬇*[b, f] L1 ≡ K1.ⓘ{I1} → 𝐔⦃f⦄ →
-                            ∀f1. f ~⊚ ⫯f1 ≡ f2 →
-                            ∃∃I2,K2. ⬇*[b, f] L2 ≡ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+                            ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+                            ∀f1. f ~⊚ ↑f1 ≘ f2 →
+                            ∃∃I2,K2. ⬇*[b, f] L2 ≘ 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
@@ -128,20 +125,19 @@ qed-.
 
 lemma lexs_drops_conf_push: ∀RN,RP.
                             ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
-                            ∀b,f,I1,K1. ⬇*[b, f] L1 ≡ K1.ⓘ{I1} → 𝐔⦃f⦄ →
-                            ∀f1. f ~⊚ ↑f1 ≡ f2 →
-                            ∃∃I2,K2. ⬇*[b, f] L2 ≡ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+                            ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+                            ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+                            ∃∃I2,K2. ⬇*[b, f] L2 ≘ 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 →
-                             ∀b,f,I2,K2. ⬇*[b, f] L2 ≡ K2.ⓘ{I2} → 𝐔⦃f⦄ →
-                             ∀f1. f ~⊚ ⫯f1 ≡ f2 →
-                             ∃∃I1,K1. ⬇*[b, f] L1 ≡ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+                             ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+                             ∀f1. f ~⊚ ↑f1 ≘ f2 →
+                             ∃∃I1,K1. ⬇*[b, f] L1 ≘ 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
@@ -149,9 +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 →
-                             ∀b,f,I2,K2. ⬇*[b, f] L2 ≡ K2.ⓘ{I2} → 𝐔⦃f⦄ →
-                             ∀f1. f ~⊚ ↑f1 ≡ f2 →
-                             ∃∃I1,K1. ⬇*[b, f] L1 ≡ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+                             ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+                             ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+                             ∃∃I1,K1. ⬇*[b, f] L1 ≘ 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
@@ -161,9 +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 →
-                             ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≡ K1 →
-                             ∀f2. f ~⊚ f1 ≡ ⫯f2 →
-                             ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≡ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+                             ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+                             ∀f2. f ~⊚ f1 ≘ ↑f2 →
+                             ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ 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
@@ -173,18 +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 →
-                             ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≡ K1 →
-                             ∀f2. f ~⊚ f1 ≡ ↑f2 →
-                             ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≡ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+                             ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+                             ∀f2. f ~⊚ f1 ≘ ⫯f2 →
+                             ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ 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: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≡ ⋆ → 𝐔⦃f1⦄ →
+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 ≡ ⋆.
+                             ∀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 //