totality of co-composition !

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / nstream_coafter.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_coafter.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_coafter.ma
new file mode 100644 (file)
index 0000000..4c141a3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_coafter.ma
@@ -0,0 +1,103 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/cocompose_2.ma".
+include "ground_2/relocation/rtmap_coafter.ma".
+
+(* RELOCATION N-STREAM ******************************************************)
+
+rec definition fun0 (n1:nat) on n1: rtmap → nat.
+* * [ | #n2 #f2 @0 ]
+#f2 cases n1 -n1 [ @0 ]
+#n1 @(⫯(fun0 n1 f2))
+defined.
+
+rec definition fun2 (n1:nat) on n1: rtmap → rtmap.
+* * [ | #n2 #f2 @(n2@f2) ]
+#f2 cases n1 -n1 [ @f2 ]
+#n1 @(fun2 n1 f2)
+defined.
+
+rec definition fun1 (n1:nat) (f1:rtmap) on n1: rtmap → rtmap.
+* * [ | #n2 #f2 @(n1@f1) ]
+#f2 cases n1 -n1 [ @f1 ]
+#n1 @(fun1 n1 f1 f2)
+defined.
+
+corec definition cocompose: rtmap → rtmap → rtmap.
+#f2 * #n1 #f1 @(seq … (fun0 n1 f2)) @(cocompose (fun2 n1 f2) (fun1 n1 f1 f2))
+defined.
+
+interpretation "functional co-composition (nstream)"
+   'CoCompose f1 f2 = (cocompose f1 f2).
+
+(* Basic properties on funs *************************************************)
+
+(* Note: we need theese since matita blocks recursive δ when ι is blocked *)  
+lemma fun0_xn: ∀f2,n1. 0 = fun0 n1 (⫯f2).
+* #n2 #f2 * //
+qed.
+
+lemma fun2_xn: ∀f2,n1. f2 = fun2 n1 (⫯f2).
+* #n2 #f2 * //
+qed.
+
+lemma fun1_xxn: ∀f2,f1,n1. fun1 n1 f1 (⫯f2) = n1@f1.
+* #n2 #f2 #f1 * //
+qed.
+
+(* Basic properies on cocompose *********************************************)
+
+lemma cocompose_rew: ∀f2,f1,n1. (fun0 n1 f2)@(fun2 n1 f2)~∘(fun1 n1 f1 f2) = f2 ~∘ (n1@f1).
+#f2 #f1 #n1 <(stream_rew … (f2~∘(n1@f1))) normalize //
+qed.
+
+(* Basic inversion lemmas on compose ****************************************)
+
+lemma cocompose_inv_ppx: ∀f2,f1,f,x. (↑f2) ~∘ (↑f1) = x@f →
+                         0 = x ∧ f2 ~∘ f1 = f.
+#f2 #f1 #f #x
+<cocompose_rew #H destruct
+normalize /2 width=1 by conj/
+qed-.
+
+lemma cocompose_inv_pnx: ∀f2,f1,f,n1,x. (↑f2) ~∘ ((⫯n1)@f1) = x@f →
+                         ∃∃n. ⫯n = x & f2 ~∘ (n1@f1) = n@f.
+#f2 #f1 #f #n1 #x
+<cocompose_rew #H destruct
+@(ex2_intro … (fun0 n1 f2)) // <cocompose_rew
+/3 width=1 by eq_f2/
+qed-.
+
+lemma cocompose_inv_nxx: ∀f2,f1,f,n1,x. (⫯f2) ~∘ (n1@f1) = x@f →
+                         0 = x ∧ f2 ~∘ (n1@f1) = f.
+#f2 #f1 #f #n1 #x
+<cocompose_rew #H destruct
+/2 width=1 by conj/
+qed-.
+
+(* Specific properties on coafter *******************************************)
+
+corec lemma coafter_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 (cocompose_inv_ppx … H) -H /3 width=7 by coafter_refl, eq_f2/
+  | #n1 #H cases (cocompose_inv_pnx … H) -H /3 width=7 by coafter_push/
+  ]
+| #n2 >next_rew #H cases (cocompose_inv_nxx … H) -H /3 width=5 by coafter_next/
+]
+qed-.
+
+theorem coafter_total: ∀f2,f1. f2 ~⊚ f1 ≡ f2 ~∘ f1.
+/2 width=1 by coafter_total_aux/ qed.