updating the dropable-related definitions with coafter ...

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
index e2f0775755c2458137fa4c6e06964fc58dd09959..0e6fcc72c86c84497b0c758e54e12a32e9fdfe2c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/relocation/rtmap_coafter.ma".
 include "basic_2/notation/relations/rdropstar_3.ma".
 include "basic_2/notation/relations/rdropstar_4.ma".
 include "basic_2/relocation/lreq.ma".
@@ -53,18 +54,18 @@ definition d_deliftable2_sn: predicate (lenv → relation term) ≝
 
 definition dropable_sn: predicate (rtmap → relation lenv) ≝
                         λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀f2,L2. R f2 L1 L2 →
-                        ∀f1. f ⊚ f1 ≡ f2 →
+                        ∀f1. f ~⊚ f1 ≡ f2 →
                         ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≡ K2.
 
 definition dropable_dx: predicate (rtmap → relation lenv) ≝
                         λR. ∀f2,L1,L2. R f2 L1 L2 →
                         ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 →  𝐔⦃f⦄ →
-                        ∀f1. f ⊚ f1 ≡ f2 → 
+                        ∀f1. f ~⊚ f1 ≡ f2 → 
                         ∃∃K1. ⬇*[b, f] L1 ≡ K1 & R f1 K1 K2.
 
 definition dedropable_sn: predicate (rtmap → relation lenv) ≝
                           λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀f1,K2. R f1 K1 K2 →
-                          ∀f2. f ⊚ f1 ≡ f2 →
+                          ∀f2. f ~⊚ f1 ≡ f2 →
                           ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
 
 (* Basic properties *********************************************************)

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..8070c5716e0c49fea362b18bb85adeee587fc7e2 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma
@@ -24,11 +24,11 @@ lemma lexs_deliftable_dropable: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_
 #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
   /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
+| #f #I #L1 #K1 #V1 #_ #IH #X #f2 #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 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:* |*: // ]
+| #f #I #L1 #K1 #V1 #W1 #HLK #HWV #IH #X #f2 #H #f1 #Hf2 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
@@ -44,11 +44,11 @@ lemma lexs_liftable_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L.
 [ #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
@@ -57,18 +57,18 @@ 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 →
+                        ∀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
@@ -86,7 +86,7 @@ lemma lexs_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
 lemma lexs_drops_conf_next: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
                             ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
                             ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
-                            ∀f1. f ⊚ ⫯f1 ≡ f2 →
+                            ∀f1. f ~⊚ ⫯f1 ≡ f2 →
                             ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{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
@@ -97,7 +97,7 @@ qed-.
 lemma lexs_drops_conf_push: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
                             ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
                             ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
-                            ∀f1. f ⊚ ↑f1 ≡ f2 →
+                            ∀f1. f ~⊚ ↑f1 ≡ f2 →
                             ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{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
@@ -108,7 +108,7 @@ 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,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
-                             ∀f1. f ⊚ ⫯f1 ≡ f2 →
+                             ∀f1. f ~⊚ ⫯f1 ≡ f2 →
                              ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{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
@@ -118,7 +118,7 @@ qed-.
 
 lemma lexs_drops_trans_push: ∀RN,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 →
+                             ∀f1. f ~⊚ ↑f1 ≡ f2 →
                              ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{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
@@ -130,7 +130,7 @@ lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. ref
                              d_liftable2 RN → d_liftable2 RP →
                              ∀f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
                              ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
-                             ∀f2. f ⊚ f1 ≡ ⫯f2 →
+                             ∀f2. f ~⊚ f1 ≡ ⫯f2 →
                              ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ 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
@@ -142,7 +142,7 @@ lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. ref
                              d_liftable2 RN → d_liftable2 RP →
                              ∀f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
                              ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
-                             ∀f2. f ⊚ f1 ≡ ↑f2 →
+                             ∀f2. f ~⊚ f1 ≡ ↑f2 →
                              ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ 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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lreq.ma
index 2ce496bd50887c1b28f525e48a0a7a476f4ddbe7..48520094936473ebd8fc46e68f56534d57f3ff4e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lreq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lreq.ma
@@ -30,7 +30,7 @@ lemma lreq_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
 (* Basic_2A1: includes: lreq_drop_trans_be *)
 lemma lreq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
                              ∀b,f,I,K2,V. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V → 𝐔⦃f⦄ →
-                             ∀f1. f ⊚ ⫯f1 ≡ f2 →
+                             ∀f1. f ~⊚ ⫯f1 ≡ f2 →
                              ∃∃K1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V & K1 ≡[f1] K2.
 #f2 #L1 #L2 #HL12 #b #f #I #K1 #V #HLK1 #Hf #f1 #Hf2
 elim (lexs_drops_trans_next … HL12 … HLK1 Hf … Hf2) -f2 -L2 -Hf
@@ -40,7 +40,7 @@ qed-.
 (* Basic_2A1: includes: lreq_drop_conf_be *)
 lemma lreq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
                             ∀b,f,I,K1,V. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V → 𝐔⦃f⦄ →
-                            ∀f1. f ⊚ ⫯f1 ≡ f2 →
+                            ∀f1. f ~⊚ ⫯f1 ≡ f2 →
                             ∃∃K2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V & K1 ≡[f1] K2.
 #f2 #L1 #L2 #HL12 #b #f #I #K1 #V #HLK1 #Hf #f1 #Hf2
 elim (lreq_drops_trans_next … (lreq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
@@ -49,7 +49,7 @@ qed-.
 
 lemma drops_lreq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
                              ∀b,f,I,L1,V. ⬇*[b,f] L1.ⓑ{I}V ≡ K1 →
-                             ∀f2. f ⊚ f1 ≡ ⫯f2 →
+                             ∀f2. f ~⊚ f1 ≡ ⫯f2 →
                              ∃∃L2. ⬇*[b,f] L2.ⓑ{I}V ≡ K2 & L1 ≡[f2] L2 & L1.ⓑ{I}V≡[f]L2.ⓑ{I}V.
 #f1 #K1 #K2 #HK12 #b #f #I #L1 #V #HLK1 #f2 #Hf2
 elim (drops_lexs_trans_next … HK12 … HLK1 … Hf2) -f1 -K1

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
index 681e07d8c701474386cf75c5abb2cea7da2a7155..47b1eec39bf521620484da8350e588a25f8303a7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
@@ -12,7 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/relocation/rtmap_coafter.ma".
 include "basic_2/relocation/drops_drops.ma".
 include "basic_2/static/frees.ma".
 

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
index 7992f334e06e86041ceefb358348c00cfe427821..e7ea1927d570991436d73f5f9e0e12c906077214 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
@@ -14,7 +14,7 @@
 
 include "ground_2/notation/relations/rcoafter_3.ma".
 include "ground_2/relocation/rtmap_sor.ma".
-include "ground_2/relocation/rtmap_istot.ma".
+include "ground_2/relocation/rtmap_after.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
@@ -213,56 +213,6 @@ lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≡ f).
 #f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
 qed-.
 
-(* Main properties **********************************************************)
-(*
-corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
-                            ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
-                            ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
-#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
-[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
-  cases (coafter_inv_xxp … Hg0 … H0) -g0
-  #f1 #f2 #Hf0 #H1 #H2
-  cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
-  #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
-  cases (coafter_inv_xxp … Hg0 … H0) -g0
-  #f1 #f2 #Hf0 #H1 #H2
-  cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
-  #f #Hf #H /3 width=7 by coafter_push/
-| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
-  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
-  [ #f1 #f2 #Hf0 #H1 #H2
-    cases (coafter_inv_nxx … Hg … H2) -g2
-    #f #Hf #H /3 width=7 by coafter_push/
-  | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
-  ]
-]
-qed-.
-
-corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
-                            ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
-                            ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
-#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
-[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
-  cases (coafter_inv_xxp … Hg0 … H0) -g0
-  #f2 #f3 #Hf0 #H2 #H3
-  cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
-  #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
-  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
-  [ #f2 #f3 #Hf0 #H2 #H3
-    cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
-    #f #Hf #H /3 width=7 by coafter_push/
-  | #f2 #Hf0 #H2
-    cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
-    #f #Hf #H /3 width=6 by coafter_next/
-  ]
-| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
-  cases (coafter_inv_nxx … Hg … H1) -g1
-  #f #Hg #H /3 width=6 by coafter_next/
-]
-qed-.
-*)
 (* Main inversion lemmas ****************************************************)
 
 corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
@@ -726,3 +676,54 @@ lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚
   /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
 ]
 qed-.
+
+(* Properties with after ****************************************************)
+(*
+corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
+                            ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
+                            ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (coafter_inv_xxp … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (coafter_inv_xxp … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by coafter_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+  [ #f1 #f2 #Hf0 #H1 #H2
+    cases (coafter_inv_nxx … Hg … H2) -g2
+    #f #Hf #H /3 width=7 by coafter_push/
+  | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
+  ]
+]
+qed-.
+
+corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
+                            ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
+                            ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (coafter_inv_xxp … Hg0 … H0) -g0
+  #f2 #f3 #Hf0 #H2 #H3
+  cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+  #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+  [ #f2 #f3 #Hf0 #H2 #H3
+    cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=7 by coafter_push/
+  | #f2 #Hf0 #H2
+    cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=6 by coafter_next/
+  ]
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+  cases (coafter_inv_nxx … Hg … H1) -g1
+  #f #Hg #H /3 width=6 by coafter_next/
+]
+qed-.
+*)