From d5216e897267e4495290ecbf08eb9f88f6815a0c Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Fri, 14 Nov 2008 20:26:02 +0000
Subject: [PATCH] test 4 luo

---
 helm/software/matita/tests/luo.ma | 110 ++++++++++++++++++++++++++++++
 1 file changed, 110 insertions(+)
 create mode 100644 helm/software/matita/tests/luo.ma

diff --git a/helm/software/matita/tests/luo.ma b/helm/software/matita/tests/luo.ma
new file mode 100644
index 000000000..1811b753c
--- /dev/null
+++ b/helm/software/matita/tests/luo.ma
@@ -0,0 +1,110 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "logic/equality.ma".
+
+inductive unit (A : Type) (a : A) : Type := mk_unit : unit A a.
+
+definition k : ∀A,a.unit A a → A := λA,a.λx:unit A a.a.
+coercion k.
+
+record monoid : Type := {
+  mC :> Type;
+  mOp : mC -> mC -> mC;
+  m1 : mC;
+  mP : ∀x:mC.x = mOp x m1
+}. 
+
+record group : Type := {
+  gC :> Type;
+  gOp : gC -> gC -> gC;
+  gInv : gC -> gC;
+  g1 : gC;
+  gP : ∀x:gC.gOp x (gInv x) = g1 
+}.
+
+(*
+record monoid_aux (x : Type) : Type := {
+  xC :> unit ? x;
+  xOp : let xC := k ?? xC in xC -> xC -> xC;
+  x1  : let xC := k ?? xC in xC;
+  xP  : let xC := k ?? xC in ∀x:xC. x = xOp x x1
+}.
+*)
+
+record monoid_aux (xC : Type) : Type := {
+  (*xC :> unit ? x;*)
+  xOp : (*let xC := k ?? xC in*) xC -> xC -> xC;
+  x1  : (*let xC := k ?? xC in*) xC;
+  xP  : (*let xC := k ?? xC in*) ∀x:xC. x = xOp x x1
+}.
+
+
+definition m_sub := λT:Type.λx:monoid_aux T.
+  mk_monoid (*k ?? (xC T x)*) T (xOp ? x) (x1 ? x) (xP ? x).
+coercion m_sub.
+
+lemma test : ∀G:group.∀M:monoid_aux (gC G).gC G = mC (m_sub ? M).
+intros; reflexivity;
+qed.
+
+record ring : Type := {
+  rG :> group;
+  rM :> monoid_aux (gC rG);
+  rP : ∀x,y,z:rG. mOp rM x (gOp rG y z) = gOp rG (mOp rM x y) (mOp rM x z)
+}.
+
+lemma prova : ∀R:ring.∀x:R. x = mOp R x (m1 R).
+intros;
+apply (mP (rM R));
+qed.
+
+include "Z/times.ma".
+
+lemma e_g : group.
+constructor 1; 
+[ apply Z;
+| apply Zplus;
+| apply (λx.-x);
+| apply OZ;
+| intros; autobatch;]
+qed.
+
+lemma e_m : monoid.
+constructor 1;
+[ apply Z;
+| apply Ztimes;
+| apply (pos O);
+| intros; autobatch;]
+qed.
+
+lemma em_1 : monoid_aux Z.
+constructor 1;
+[ apply Ztimes;
+| apply (pos O);
+| intros; autobatch;]
+qed.
+
+lemma e_r : ring.
+constructor 1;
+[ apply e_g; | apply em_1;| intros; unfold e_g; unfold em_1; simplify; autobatch;]
+qed.
+
+lemma test1 : ∀x:Z.x = x * pos O.
+intros; apply (mP e_r x);
+qed.
+  
+
+
+ 
-- 
2.39.2