X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Fnstream_after.ma;h=ca3ad7bc7c44535ed250d434ea40d6549336f835;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=ae7add375539397cb2ef84e0aab7fbadab3b4af8;hpb=6acee1cf296163fee832b112c96b6624253aee06;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
index ae7add375..ca3ad7bc7 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
@@ -12,426 +12,147 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/notation/relations/rafter_3.ma".
-include "ground_2/lib/streams_hdtl.ma".
-include "ground_2/relocation/nstream_at.ma".
+include "ground_2/relocation/nstream_istot.ma".
+include "ground_2/relocation/rtmap_after.ma".
 
 (* RELOCATION N-STREAM ******************************************************)
 
-let corec compose: rtmap → rtmap → rtmap ≝ ?.
-#f1 * #b2 #f2 @(seq … (f1@❴b2❵)) @(compose ? f2) -compose -f2
-@(tln … (⫯b2) f1)
-qed.
+corec definition compose: rtmap → rtmap → rtmap.
+#f2 * #n1 #f1 @(seq … (f2@❨n1❩)) @(compose ? f1) -compose -f1
+@(⫰*[↑n1] f2)
+defined.
 
 interpretation "functional composition (nstream)"
-   'compose f1 f2 = (compose f1 f2).
-
-coinductive after: relation3 rtmap rtmap rtmap ≝
-| after_zero: ∀f1,f2,f,b1,b2,b.
-              after f1 f2 f →
-              b1 = 0 → b2 = 0 → b = 0 →
-              after (b1@f1) (b2@f2) (b@f)
-| after_skip: ∀f1,f2,f,b1,b2,b,a2,a.
-              after f1 (a2@f2) (a@f) →
-              b1 = 0 → b2 = ⫯a2 → b = ⫯a →
-              after (b1@f1) (b2@f2) (b@f)
-| after_drop: ∀f1,f2,f,b1,b,a1,a.
-              after (a1@f1) f2 (a@f) →
-              b1 = ⫯a1 → b = ⫯a →
-              after (b1@f1) f2 (b@f)
-.
-
-interpretation "relational composition (nstream)"
-   'RAfter f1 f2 f = (after f1 f2 f).
+   'compose f2 f1 = (compose f2 f1).
 
 (* Basic properies on compose ***********************************************)
 
-lemma compose_unfold: ∀f1,f2,a2. f1∘(a2@f2) = f1@❴a2❵@tln … (⫯a2) f1∘f2.
-#f1 #f2 #a2 >(stream_expand … (f1∘(a2@f2))) normalize //
+lemma compose_rew: ∀f2,f1,n1. f2@❨n1❩⨮(⫰*[↑n1]f2)∘f1 = f2∘(n1⨮f1).
+#f2 #f1 #n1 <(stream_rew … (f2∘(n1⨮f1))) normalize //
 qed.
 
-lemma compose_drop: ∀f1,f2,f,a1,a. (a1@f1)∘f2 = a@f → (⫯a1@f1)∘f2 = ⫯a@f.
-#f1 * #a2 #f2 #f #a1 #a >compose_unfold >compose_unfold
-#H destruct normalize //
+lemma compose_next: ∀f2,f1,f. f2∘f1 = f → (↑f2)∘f1 = ↑f.
+#f2 * #n1 #f1 #f <compose_rew <compose_rew
+* -f <tls_S1 /2 width=1 by eq_f2/
 qed.
 
 (* Basic inversion lemmas on compose ****************************************)
 
-lemma compose_inv_unfold: ∀f1,f2,f,a2,a. f1∘(a2@f2) = a@f →
-                          f1@❴a2❵ = a ∧ tln … (⫯a2) f1∘f2 = f.
-#f1 #f2 #f #a2 #a >(stream_expand … (f1∘(a2@f2))) normalize
+lemma compose_inv_rew: ∀f2,f1,f,n1,n. f2∘(n1⨮f1) = n⨮f →
+                       f2@❨n1❩ = n ∧ (⫰*[↑n1]f2)∘f1 = f.
+#f2 #f1 #f #n1 #n <(stream_rew … (f2∘(n1⨮f1))) normalize
 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_O2: ∀f1,f2,f,a1,a. (a1@f1)∘(O@f2) = a@f →
-                      a = a1 ∧ f1∘f2 = f.
-#f1 #f2 #f #a1 #a >compose_unfold
+lemma compose_inv_O2: ∀f2,f1,f,n2,n. (n2⨮f2)∘(⫯f1) = n⨮f →
+                      n2 = n ∧ f2∘f1 = f.
+#f2 #f1 #f #n2 #n <compose_rew
 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_S2: ∀f1,f2,f,a1,a2,a. (a1@f1)∘(⫯a2@f2) = a@f →
-                      a = ⫯(a1+f1@❴a2❵) ∧ f1∘(a2@f2) = f1@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
-#H destruct /2 width=1 by conj/
+lemma compose_inv_S2: ∀f2,f1,f,n2,n1,n. (n2⨮f2)∘(↑n1⨮f1) = n⨮f →
+                      ↑(n2+f2@❨n1❩) = n ∧ f2∘(n1⨮f1) = f2@❨n1❩⨮f.
+#f2 #f1 #f #n2 #n1 #n <compose_rew
+#H destruct <tls_S1 /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_S1: ∀f1,f2,f,a1,a2,a. (⫯a1@f1)∘(a2@f2) = a@f →
-                      a = ⫯((a1@f1)@❴a2❵) ∧ (a1@f1)∘(a2@f2) = (a1@f1)@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
-#H destruct /2 width=1 by conj/
+lemma compose_inv_S1: ∀f2,f1,f,n1,n. (↑f2)∘(n1⨮f1) = n⨮f →
+                      ↑(f2@❨n1❩) = n ∧ f2∘(n1⨮f1) = f2@❨n1❩⨮f.
+#f2 #f1 #f #n1 #n <compose_rew
+#H destruct <tls_S1 /2 width=1 by conj/
 qed-.
 
-(* Basic properties on after ************************************************)
+(* Specific properties on after *********************************************)
 
-lemma after_O2: ∀f1,f2,f. f1 ⊚ f2 ≡ f →
-                ∀b. b@f1 ⊚ O@f2 ≡ b@f.
-#f1 #f2 #f #Ht #b elim b -b /2 width=5 by after_drop, after_zero/
+lemma after_O2: ∀f2,f1,f. f2 ⊚ f1 ≘ f →
+                ∀n. n⨮f2 ⊚ ⫯f1 ≘ n⨮f.
+#f2 #f1 #f #Hf #n elim n -n /2 width=7 by after_refl, after_next/
 qed.
 
-lemma after_S2: ∀f1,f2,f,b2,b. f1 ⊚ b2@f2 ≡ b@f →
-                ∀b1. b1@f1 ⊚ ⫯b2@f2 ≡ ⫯(b1+b)@f.
-#f1 #f2 #f #b2 #b #Ht #b1 elim b1 -b1 /2 width=5 by after_drop, after_skip/
+lemma after_S2: ∀f2,f1,f,n1,n. f2 ⊚ n1⨮f1 ≘ n⨮f →
+                ∀n2. n2⨮f2 ⊚ ↑n1⨮f1 ≘ ↑(n2+n)⨮f.
+#f2 #f1 #f #n1 #n #Hf #n2 elim n2 -n2 /2 width=7 by after_next, after_push/
 qed.
 
-lemma after_apply: ∀b2,f1,f2,f. (tln … (⫯b2) f1) ⊚ f2 ≡ f → f1 ⊚ b2@f2 ≡ f1@❴b2❵@f.
-#b2 elim b2 -b2
+lemma after_apply: ∀n1,f2,f1,f. (⫰*[↑n1] f2) ⊚ f1 ≘ f → f2 ⊚ n1⨮f1 ≘ f2@❨n1❩⨮f.
+#n1 elim n1 -n1
 [ * /2 width=1 by after_O2/
-| #b2 #IH * /3 width=1 by after_S2/
+| #n1 #IH * /3 width=1 by after_S2/
 ]
 qed-.
 
-let corec after_total_aux: ∀f1,f2,f. f1 ∘ f2 = f → f1 ⊚ f2 ≡ f ≝ ?.
-* #a1 #f1 * #a2 #f2 * #a #f cases a1 -a1
-[ cases a2 -a2
-  [ #H cases (compose_inv_O2 … H) -H
-    /3 width=1 by after_zero/
-  | #a2 #H cases (compose_inv_S2 … H) -H
-    /3 width=5 by after_skip, eq_f/
+corec lemma after_total_aux: ∀f2,f1,f. f2 ∘ f1 = f → f2 ⊚ f1 ≘ f.
+* #n2 #f2 * #n1 #f1 * #n #f cases n2 -n2
+[ cases n1 -n1
+  [ #H cases (compose_inv_O2 … H) -H /3 width=7 by after_refl, eq_f2/
+  | #n1 #H cases (compose_inv_S2 … H) -H * -n /3 width=7 by after_push/
   ]
-| #a1 #H cases (compose_inv_S1 … H) -H
-  /3 width=5 by after_drop, eq_f/
+| #n2 >next_rew #H cases (compose_inv_S1 … H) -H * -n /3 width=5 by after_next/
 ]
 qed-.
 
-theorem after_total: ∀f2,f1. f1 ⊚ f2 ≡ f1 ∘ f2.
+theorem after_total: ∀f1,f2. f2 ⊚ f1 ≘ f2 ∘ f1.
 /2 width=1 by after_total_aux/ qed.
 
-(* Basic inversion lemmas on after ******************************************)
-
-fact after_inv_O1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1. f1 = 0@g1 →
-                       (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
-                       ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
-| #b2 #b #a2 #a #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=7 by ex3_4_intro, or_intror/
-| #b #a1 #a #_ #H1 #H3 #g1 #H destruct
-]
-qed-.
-
-fact after_inv_O1_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 →
-                        (∧∧ f1 ⊚ f2 ≡ f & b2 = 0 & b = 0) ∨
-                        ∃∃a2,a. f1 ⊚ a2@f2 ≡ a@f & b2 = ⫯a2 & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H elim (after_inv_O1_aux … Ht) -Ht [4: // |2: skip ] *
-[ #g2 #g #Hu #H1 #H2 destruct /3 width=1 by and3_intro, or_introl/
-| #g2 #g #a2 #a #Hu #H1 #H2 destruct /3 width=5 by ex3_2_intro, or_intror/
-]
-qed-.
-
-lemma after_inv_O1: ∀g1,f2,f. 0@g1 ⊚ f2 ≡ f →
-                    (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
-                    ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-/2 width=3 by after_inv_O1_aux/ qed-.
-
-fact after_inv_zero_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → b2 = 0 →
-                          f1 ⊚ f2 ≡ f ∧ b = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ /2 width=1 by conj/
-| #a1 #a2 #_ #H0 destruct
-]
-qed-.
-
-lemma after_inv_zero: ∀g1,g2,f. 0@g1 ⊚ 0@g2 ≡ f →
-                      ∃∃g. g1 ⊚ g2 ≡ g & f = 0@g.
-#g1 #g2 #f #H elim (after_inv_O1 … H) -H *
-[ #x2 #g #Hu #H1 #H2 destruct /2 width=3 by ex2_intro/
-| #x2 #g #a2 #a #Hu #H destruct
-]
-qed-.
-
-fact after_inv_skip_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → ∀a2. b2 = ⫯a2 →
-                          ∃∃a. f1 ⊚ a2@f2 ≡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #a2 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ #_ #H0 destruct
-| #x2 #x #H #H0 #H1 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma after_inv_skip: ∀g1,g2,f,b2. 0@g1 ⊚ ⫯b2@g2 ≡ f →
-                      ∃∃g,b. g1 ⊚ b2@g2 ≡ b@g & f = ⫯b@g.
-#g1 #g2 * #b #f #b2 #Ht elim (after_inv_skip_aux2 … Ht) [2,4: // |3: skip ] -Ht
-#a #Ht #H destruct /2 width=4 by ex2_2_intro/
-qed-.
-
-fact after_inv_S1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1,b1. f1 = ⫯b1@g1 →
-                       ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b #a1 #a #Ht #H1 #H3 #g1 #x1 #H destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-fact after_inv_S1_aux2: ∀f1,f2,f,b1,b. b1@f1 ⊚ f2 ≡ b@f → ∀a1. b1 = ⫯a1 →
-                        ∃∃a. a1@f1 ⊚ f2 ≡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b #Ht #a #H elim (after_inv_S1_aux … Ht) -Ht [4: // |2,3: skip ]
-#g #x #Hu #H0 destruct /2 width=3 by ex2_intro/ 
-qed-.
-
-lemma after_inv_S1: ∀g1,f2,f,b1. ⫯b1@g1 ⊚ f2 ≡ f →
-                    ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-/2 width=3 by after_inv_S1_aux/ qed-.
-
-fact after_inv_drop_aux2: ∀f1,f2,f,a1,a. a1@f1 ⊚ f2 ≡ a@f → ∀b1,b. a1 = ⫯b1 → a = ⫯b →
-                          b1@f1 ⊚ f2 ≡ b@f.
-#f1 #f2 #f #a1 #a #Ht #b1 #b #H1 #H elim (after_inv_S1_aux2 … Ht … H1) -a1
-#x #Ht #Hx destruct //
-qed-.
-
-lemma after_inv_drop: ∀f1,f2,f,b1,b. ⫯b1@f1 ⊚ f2 ≡ ⫯b@f → b1@f1 ⊚ f2 ≡ b@f.
-/2 width=5 by after_inv_drop_aux2/ qed-.
-
-fact after_inv_O3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g. f = 0@g →
-                        ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g #H destruct /2 width=5 by ex3_2_intro/
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g #H destruct
-| #b #a1 #a #_ #H1 #H3 #g #H destruct
-]
-qed-.
-
-fact after_inv_O3_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b = 0 →
-                        ∧∧ f1 ⊚ f2 ≡ f & b1 = 0 & b2 = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 elim (after_inv_O3_aux1 … Ht) [2: // |3: skip ] -b
-#g1 #g2 #Ht #H1 #H2 destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma after_inv_O3: ∀f1,f2,g. f1 ⊚ f2 ≡ 0@g →
-                    ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-/2 width=3 by after_inv_O3_aux1/ qed-.
-
-fact after_inv_S3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g,b. f = ⫯b@g →
-                        (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
-                        ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g #a #H destruct
-| #b2 #b #a2 #a #HT #H1 #H2 #H3 #g #x #H destruct /3 width=6 by ex3_3_intro, or_introl/
-| #b #a1 #a #HT #H1 #H3 #g #x #H destruct /3 width=4 by ex2_2_intro, or_intror/
-]
-qed-.
-
-fact after_inv_S3_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → ∀b. a = ⫯b →
-                        (∃∃b2. f1 ⊚ b2@f2 ≡ b@f & a1 = 0 & a2 = ⫯b2) ∨
-                        ∃∃b1. b1@f1 ⊚ a2@f2 ≡ b@f & a1 = ⫯b1.
-#f1 #f2 #f #a1 #a2 #a #Ht #b #H elim (after_inv_S3_aux1 … Ht) [3: // |4,5: skip ] -a *
-[ #g1 #g2 #b2 #Ht #H1 #H2 destruct /3 width=3 by ex3_intro, or_introl/
-| #g1 #b1 #Ht #H1 destruct /3 width=3 by ex2_intro, or_intror/
-]
-qed-.
-
-lemma after_inv_S3: ∀f1,f2,g,b. f1 ⊚ f2 ≡ ⫯b@g →
-                    (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
-                    ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-/2 width=3 by after_inv_S3_aux1/ qed-.
-
-(* Advanced inversion lemmas on after ***************************************)
-
-fact after_inv_O2_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → a2 = 0 →
-                         a1 = a ∧ f1 ⊚ f2 ≡ f.
-#f1 #f2 #f #a1 #a2 elim a1 -a1
-[ #a #H #H2 elim (after_inv_zero_aux2 … H … H2) -a2 /2 width=1 by conj/
-| #a1 #IH #a #H #H2 elim (after_inv_S1_aux2 … H) -H [3: // |2: skip ]
-  #b #H #H1 elim (IH … H) // -a2
-  #H2 destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma after_inv_O2: ∀f1,g2,f. f1 ⊚ 0@g2 ≡ f →
-                    ∃∃g1,g,a. f1 = a@g1 & f = a@g & g1 ⊚ g2 ≡ g.
-* #a1 #f1 #f2 * #a #f #H elim (after_inv_O2_aux2 … H) -H //
-/2 width=6 by ex3_3_intro/
-qed-.
+(* Specific inversion lemmas on after ***************************************)
 
-lemma after_inv_const: ∀a,f1,b2,g2,f. a@f1 ⊚ b2@g2 ≡ a@f → b2 = 0.
-#a elim a -a
-[ #f1 #b2 #g2 #f #H elim (after_inv_O3 … H) -H
-  #g1 #x2 #_ #_ #H destruct //
-| #a #IH #f1 #b2 #g2 #f #H elim (after_inv_S1 … H) -H
-  #x #b #Hx #H destruct >(IH … Hx) -f1 -g2 -x -b2 -b //
+lemma after_inv_xpx: ∀f2,g2,f,n2,n. n2⨮f2 ⊚ g2 ≘ n⨮f → ∀f1. ⫯f1 = g2 →
+                     f2 ⊚ f1 ≘ f ∧ n2 = n.
+#f2 #g2 #f #n2 elim n2 -n2
+[ #n #Hf #f1 #H2 elim (after_inv_ppx … Hf … H2) -g2 [2,3: // ]
+  #g #Hf #H elim (push_inv_seq_dx … H) -H destruct /2 width=1 by conj/
+| #n2 #IH #n #Hf #f1 #H2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+  #g1 #Hg #H1 elim (next_inv_seq_dx … H1) -H1
+  #x #Hx #H destruct elim (IH … Hg) [2,3: // ] -IH -Hg
+  #H destruct /2 width=1 by conj/
 ]
 qed-.
 
-lemma after_inv_S2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ ⫯a2@f2 ≡ a@f → ∀b. a = ⫯(a1+b) →
-                    f1 ⊚ a2@f2 ≡ b@f.
-#f1 #f2 #f #a1 elim a1 -a1
-[ #a2 #a #Ht #b #Hb
-  elim (after_inv_skip_aux2 … Ht) -Ht [3,4: // |2: skip ]
-  #c #Ht #Hc destruct //
-| #a1 #IH #a2 #a #Ht #b #Hb
-  lapply (after_inv_drop_aux2 … Ht … Hb) -a [ // | skip ]
-  /2 width=3 by/
-]
-qed-.
-
-(* Forward lemmas on application ********************************************)
-
-lemma after_at_fwd: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
-                    ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
-#f #i1 #i #H elim H -f -i1 -i
-[ #f #f2 #f1 #H elim (after_inv_O3 … H) -H
-  /2 width=3 by at_zero, ex2_intro/
-| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_O3 … H) -H
-  #g2 #g1 #Hu #H1 #H2 destruct elim (IH … Hu) -f
-  /3 width=3 by at_S1, ex2_intro/
-| #f #b #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_S3 … H) -H *
-  [ #g2 #g1 #b2 #Hu #H1 #H2 destruct elim (IH … Hu) -f -b
-    /3 width=3 by at_S1, at_lift, ex2_intro/
-  | #g1 #b1 #Hu #H destruct elim (IH … Hu) -f -b
-    /3 width=3 by at_lift, ex2_intro/
-  ]
+lemma after_inv_xnx: ∀f2,g2,f,n2,n. n2⨮f2 ⊚ g2 ≘ n⨮f → ∀f1. ↑f1 = g2 →
+                     ∃∃m. f2 ⊚ f1 ≘ m⨮f & ↑(n2+m) = n.
+#f2 #g2 #f #n2 elim n2 -n2
+[ #n #Hf #f1 #H2 elim (after_inv_pnx … Hf … H2) -g2 [2,3: // ]
+  #g #Hf #H elim (next_inv_seq_dx … H) -H
+  #x #Hx #Hg destruct /2 width=3 by ex2_intro/
+| #n2 #IH #n #Hf #f1 #H2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+  #g #Hg #H elim (next_inv_seq_dx … H) -H
+  #x #Hx #H destruct elim (IH … Hg) -IH -Hg [2,3: // ]
+  #m #Hf #Hm destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2,f. f2 ⊚ f1 ≡ f →
-                     ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
-#f1 #i1 #i2 #H elim H -f1 -i1 -i2
-[ #f1 #f2 #f #H elim (after_inv_O2 … H) -H /2 width=3 by ex2_intro/
-| #f1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
-  [ #f2 #f #H elim (after_inv_zero … H) -H
-    #g #Hu #H destruct elim (IH … Hu) -f1
-    /3 width=3 by at_S1, at_skip, ex2_intro/
-  | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
-    #g #b #Hu #H destruct elim (IH … Hu) -f1
-    /3 width=3 by at_lift, ex2_intro/
-  ]
-| #f1 #b1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
-  [ #f2 #f #H elim (after_inv_skip … H) -H
-    #g #a #Hu #H destruct elim (IH … Hu) -f1 -b1
-    /3 width=3 by at_S1, at_lift, ex2_intro/
-  | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
-    #g #b #Hu #H destruct elim (IH … Hu) -f1 -b1
-    /3 width=3 by at_lift, ex2_intro/
-  ]
+lemma after_inv_const: ∀f2,f1,f,n1,n. n⨮f2 ⊚ n1⨮f1 ≘ n⨮f → f2 ⊚ f1 ≘ f ∧ 0 = n1.
+#f2 #f1 #f #n1 #n elim n -n
+[ #H elim (after_inv_pxp … H) -H [ |*: // ]
+  #g2 #Hf #H elim (push_inv_seq_dx … H) -H /2 width=1 by conj/
+| #n #IH #H lapply (after_inv_nxn … H ????) -H /2 width=5 by/
 ]
 qed-.
 
-lemma after_fwd_at: ∀f1,f2,i1,i2,i. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
-                    ∀f. f2 ⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
-#f1 #f2 #i1 #i2 #i #Hi1 #Hi2 #f #Ht elim (after_at1_fwd … Hi1 … Ht) -f1
-#j #H #Hj >(at_mono … H … Hi2) -i2 //
-qed-.
-
-lemma after_fwd_at1: ∀f2,f,i1,i2,i. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
-                     ∀f1. f2 ⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
-#f2 #f #i1 #i2 #i #Hi1 #Hi2 #f1 #Ht elim (after_at_fwd … Hi1 … Ht) -f
-#j1 #Hij1 #H >(at_inj … Hi2 … H) -i //
-qed-.
-
-lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
-                     ∀f2. f2 ⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
-#f #i1 #i #H elim H -f -i1 -i
-[ #f #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
-  #g2 #g1 #_ #H1 #H2 destruct >(at_inv_OOx … Ht1) -f -g1 -i2 //
-| #f #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
-  #g2 #g1 #Hu #H1 #H2 destruct elim (at_inv_SOx … Ht1) -Ht1
-  /3 width=3 by at_skip/
-| #f #b #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_S3 … H) -H *
-  [ #g2 #g1 #a1 #Hu #H1 #H2 destruct elim (at_inv_xSx … Ht1) -Ht1
-    /3 width=3 by at_skip/
-  | #g2 #a2 #Hu #H destruct /3 width=3 by at_lift/
-  ]
-]
-qed-.
+lemma after_inv_total: ∀f2,f1,f. f2 ⊚ f1 ≘ f → f2 ∘ f1 ≡ f.
+/2 width=4 by after_mono/ qed-.
 
-(* Advanced forward lemmas on after *****************************************)
+(* Specific forward lemmas on after *****************************************)
 
-lemma after_fwd_hd: ∀f1,f2,f,a2,a. f1 ⊚ a2@f2 ≡ a@f → a = f1@❴a2❵.
-#f1 #f2 #f #a2 #a #Ht lapply (after_fwd_at … 0 … Ht) -Ht [4: // | // |2,3: skip ]
+lemma after_fwd_hd: ∀f2,f1,f,n1,n. f2 ⊚ n1⨮f1 ≘ n⨮f → f2@❨n1❩ = n.
+#f2 #f1 #f #n1 #n #H lapply (after_fwd_at ? n1 0 … H) -H [1,2,3: // ]
 /3 width=2 by at_inv_O1, sym_eq/
 qed-.
 
-lemma after_fwd_tl: ∀f,f2,a2,f1,a1,a. a1@f1 ⊚ a2@f2 ≡ a@f →
-                    tln … a2 f1 ⊚ f2 ≡ f.
-#f #f2 #a2 elim a2 -a2
-[ #f1 #a1 #a #Ht elim (after_inv_O2_aux2 … Ht) -Ht //
-| #a2 #IH * #b1 #f1 #a1 #a #Ht
-  lapply (after_fwd_hd … Ht) #Ha
-  lapply (after_inv_S2 … Ht … Ha) -a
-  /2 width=3 by/
-]
-qed-.
-
-lemma after_inv_apply: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f →
-                       a = (a1@f1)@❴a2❵ ∧ tln … a2 f1 ⊚ f2 ≡ f.
-/3 width=3 by  after_fwd_tl, after_fwd_hd, conj/ qed-.
-
-(* Main properties on after *************************************************)
-
-let corec after_trans1: ∀f1,f2,f0. f1 ⊚ f2 ≡ f0 →
-                        ∀f3,f4. f0 ⊚ f3 ≡ f4 →
-                        ∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4 ≝ ?.
-#f1 #f2 #f0 * -f1 -f2 -f0 #f1 #f2 #f0 #b1 [1,2: #b2 ] #b0
-[ #Ht0 #H1 #H2 #H0 * #b3 #f3 * #b4 #f4 #Ht4 * #b #f #Ht
-  cases (after_inv_O1_aux2 … Ht4 H0) -Ht4 -H0 *
-  [ #Ht4 #H3 #H4 cases (after_inv_zero_aux2 … Ht H2 H3) -Ht -H2 -H3
-    #Ht #H /3 width=6 by after_zero/
-  | #a0 #a4 #Ht4 #H3 #H4 cases (after_inv_skip_aux2 … Ht H2 … H3) -Ht -H2 -H3
-    #a #Ht3 #H /3 width=6 by after_skip/
-  ]
-| #a2 #a0 #Ht0 #H1 #H2 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
-  #a4 #Ht4 #H4 * #b #f #H cases (after_inv_S1_aux2 … H … H2) -H -H2
-  #a #Ht3 #H /3 width=6 by after_skip/
-| #a1 #a0 #Ht0 #H1 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
-  #a4 #Ht4 #H4 * #b #f #Ht /3 width=6 by after_drop/
-]
-qed-.
-
-let corec after_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 #b1 [1,2: #b0 ] #b4
-[ #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_O3_aux2 … Ht0 H0) -b0
-  #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
-  #Ht #H /3 width=6 by after_zero/
-| #a0 #a4 #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_S3_aux2 … Ht0 … H0) -b0 *
-  [ #a3 #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
-    #Ht #H /3 width=6 by after_skip/
-  | #a2 #Ht0 #H2 cases (after_inv_skip_aux2 … Ht H1 … H2) -b1 -b2
-    #a #Ht #H /3 width=6 by after_drop/
-  ]
-| #a1 #a4 #Ht4 #H1 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_S1_aux2 … Ht … H1) -b1
-  #a #Ht #H /3 width=6 by after_drop/
+lemma after_fwd_tls: ∀f,f1,n1,f2,n2,n. n2⨮f2 ⊚ n1⨮f1 ≘ n⨮f →
+                     (⫰*[n1]f2) ⊚ f1 ≘ f.
+#f #f1 #n1 elim n1 -n1
+[ #f2 #n2 #n #H elim (after_inv_xpx … H) -H //
+| #n1 #IH * #m1 #f2 #n2 #n #H elim (after_inv_xnx … H) -H [2,3: // ]
+  #m #Hm #H destruct /2 width=3 by/
 ]
 qed-.
 
-let corec after_mono: ∀f1,f2,x. f1 ⊚ f2 ≡ x → ∀y. f1 ⊚ f2 ≡ y → x ≐ y ≝ ?.
-* #a1 #f1 * #a2 #f2 * #c #x #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd #Hy
-/3 width=4 by eq_seq/
-qed-.
+lemma after_inv_apply: ∀f2,f1,f,n2,n1,n. n2⨮f2 ⊚ n1⨮f1 ≘ n⨮f →
+                       (n2⨮f2)@❨n1❩ = n ∧ (⫰*[n1]f2) ⊚ f1 ≘ f.
+/3 width=3 by after_fwd_tls, after_fwd_hd, conj/ qed-.
 
-let corec after_inj: ∀f1,x,f. f1 ⊚ x ≡ f → ∀y. f1 ⊚ y ≡ f → x ≐ y ≝ ?.
-* #a1 #f1 * #c #x * #a #f #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd
-cases (apply_inj_aux … Hc Hd) //
-#Hy -a -d /3 width=4 by eq_seq/
-qed-.
+(* Properties on apply ******************************************************)
 
-(* Main inversion lemmas on after *******************************************)
-
-theorem after_inv_total: ∀f1,f2,f. f1 ⊚ f2 ≡ f → f1 ∘ f2 ≐ f.
-/2 width=4 by after_mono/ qed-.
+lemma compose_apply (f2) (f1) (i): f2@❨f1@❨i❩❩ = (f2∘f1)@❨i❩.
+/4 width=6 by after_fwd_at, at_inv_total, sym_eq/ qed.