snopshot (working one!)

[helm.git] / matita / tests / push_pop_status.ma

diff --git a/matita/tests/push_pop_status.ma b/matita/tests/push_pop_status.ma
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+
+(* XXX coercions *)
+axiom A: Type.
+axiom B: Type.
+axiom cAB: A -> B.
+coercion cic:/matita/tests/push_pop_status/cAB.con.
+
+inductive eq (A:Type) (x:A) : A \to Prop \def
+    refl_eq : eq A x x.
+
+(* XXX notation *)
+notation "hvbox(a break === b)" non associative with precedence 45 for @{ 'eqqq $a $b }.
+interpretation "test" 'eqqq x y = (cic:/matita/tests/push_pop_status/eq.ind#xpointer(1/1) _ x y).
+
+theorem eq_rect':
+ \forall A. \forall x:A. \forall P: \forall y:A. x===y \to Type.
+  P ? (refl_eq ? x) \to \forall y:A. \forall p:x===y. P y p.
+ intros.
+ exact
+  (match p1 return \lambda y. \lambda p.P y p with
+    [refl_eq \Rightarrow p]).
+qed.
+ 
+lemma sym_eq : \forall A:Type.\forall x,y:A. x===y  \to y===x.
+intros.elim H. apply refl_eq.
+qed.
+
+lemma trans_eq : \forall A:Type.\forall x,y,z:A. x===y  \to y===z \to x===z.
+intros.elim H1.assumption.
+qed.
+
+theorem eq_elim_r:
+ \forall A:Type.\forall x:A. \forall P: A \to Prop.
+   P x \to \forall y:A. y===x \to P y.
+intros. elim (sym_eq ? ? ? H1).assumption.
+qed.
+
+theorem eq_elim_r':
+ \forall A:Type.\forall x:A. \forall P: A \to Set.
+   P x \to \forall y:A. y===x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
+theorem eq_elim_r'':
+ \forall A:Type.\forall x:A. \forall P: A \to Type.
+   P x \to \forall y:A. y===x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
+theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
+\forall x,y:A. x===y \to f x === f y.
+intros.elim H.apply refl_eq.
+qed.
+
+theorem eq_f': \forall  A,B:Type.\forall f:A\to B.
+\forall x,y:A. x===y \to f y === f x.
+intros.elim H.apply refl_eq.
+qed.
+
+(* XXX defaults *)
+default "equality"
+ cic:/matita/tests/push_pop_status/eq.ind
+ cic:/matita/tests/push_pop_status/sym_eq.con
+ cic:/matita/tests/push_pop_status/trans_eq.con
+ cic:/matita/tests/push_pop_status/eq_ind.con
+ cic:/matita/tests/push_pop_status/eq_elim_r.con
+ cic:/matita/tests/push_pop_status/eq_rec.con
+ cic:/matita/tests/push_pop_status/eq_elim_r'.con
+ cic:/matita/tests/push_pop_status/eq_rect.con
+ cic:/matita/tests/push_pop_status/eq_elim_r''.con
+ cic:/matita/tests/push_pop_status/eq_f.con
+ cic:/matita/tests/push_pop_status/eq_f'.con. (* \x.sym (eq_f x) *)
+ 
+include "push_pop_status_aux1.ma".
+(* include "push_pop_status_aux2.ma". *)
+
+(* XXX default *)
+theorem prova: \forall x:A. eq A x x.
+intros. reflexivity.
+qed.
+
+(* XXX notation *)
+theorem prova1: \forall x:A. x === x.
+intros. apply refl_eq.
+qed.
+
+(* XXX coercion *)
+theorem pippo: A -> B.
+intro a.
+apply (a : B).
+qed.
+
+definition X := b.
+
+