freescale porting, work in progress

[helm.git] / helm / software / matita / contribs / ng_assembly / num / quatern_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma b/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma
new file mode 100755 (executable)
index 0000000..66a75cc
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+++ b/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma
@@ -0,0 +1,115 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(*                          Progetto FreeScale                            *)
+(*                                                                        *)
+(*   Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it                   *)
+(*     Cosimo Oliboni, oliboni@cs.unibo.it                                *)
+(*                                                                        *)
+(* ********************************************************************** *)
+
+include "num/quatern.ma".
+include "num/bool_lemmas.ma".
+
+(* ********** *)
+(* QUATERNARI *)
+(* ********** *)
+
+ndefinition quatern_destruct_aux ≝
+Πn1,n2:quatern.ΠP:Prop.n1 = n2 →
+ match n1 with
+  [ q0 ⇒ match n2 with [ q0 ⇒ P → P | _ ⇒ P ]
+  | q1 ⇒ match n2 with [ q1 ⇒ P → P | _ ⇒ P ]
+  | q2 ⇒ match n2 with [ q2 ⇒ P → P | _ ⇒ P ]
+  | q3 ⇒ match n2 with [ q3 ⇒ P → P | _ ⇒ P ]
+  ].
+
+ndefinition quatern_destruct : quatern_destruct_aux.
+ #n1; #n2; #P;
+ nelim n1;
+ ##[ ##1: nelim n2; nnormalize; #H;
+          ##[ ##1: napply (λx:P.x)
+          ##| ##*: napply False_ind;
+                   nchange with (match q0 with [ q0 ⇒ False | _ ⇒ True ]);
+                   nrewrite > H; nnormalize; napply I
+          ##]
+ ##| ##2: nelim n2; nnormalize; #H;
+          ##[ ##2: napply (λx:P.x)
+          ##| ##*: napply False_ind;
+                   nchange with (match q1 with [ q1 ⇒ False | _ ⇒ True ]);
+                   nrewrite > H; nnormalize; napply I
+          ##]
+ ##| ##3: nelim n2; nnormalize; #H;
+          ##[ ##3: napply (λx:P.x)
+          ##| ##*: napply False_ind;
+                   nchange with (match q2 with [ q2 ⇒ False | _ ⇒ True ]);
+                   nrewrite > H; nnormalize; napply I
+          ##]
+ ##| ##4: nelim n2; nnormalize; #H;
+          ##[ ##4: napply (λx:P.x)
+          ##| ##*: napply False_ind;
+                   nchange with (match q3 with [ q3 ⇒ False | _ ⇒ True ]);
+                   nrewrite > H; nnormalize; napply I
+          ##]
+ ##]
+nqed.
+
+nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
+ #n1; #n2;
+ nelim n1;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,6,11,16: #H; napply refl_eq
+ ##| ##*: #H; napply (bool_destruct … H)
+ ##]
+nqed.
+
+nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,6,11,16: #H; napply refl_eq
+ ##| ##*: #H; napply (quatern_destruct … H)
+ ##]
+nqed.
+
+nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,6,11,16: #H; napply (bool_destruct … H)
+ ##| ##*: #H; #H1; napply (quatern_destruct … H1)
+ ##]
+nqed.
+
+nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,6,11,16: #H; napply False_ind; napply (H (refl_eq …))
+ ##| ##*: #H; napply refl_eq
+ ##]
+nqed.