+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/tests/".
+
+
inductive True: Prop \def
I : True.
inductive False: Prop \def .
definition Not: Prop \to Prop \def
-\lambda A:Prop. (A \to False).
+\lambda A. (A \to False).
theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-intro.cut False.elim Hcut.apply H1.assumption.
+intros. elim (H1 H).
qed.
inductive And (A,B:Prop) : Prop \def
conj : A \to B \to (And A B).
theorem proj1: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
qed.
theorem proj2: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
qed.
inductive Or (A,B:Prop) : Prop \def
refl_equal : eq A x x.
theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y \to eq A y x.
-intro. elim H. apply refl_equal.
+intros. elim H. apply refl_equal.
qed.
theorem trans_eq : \forall A:Type.
\forall x,y,z:A. eq A x y \to eq A y z \to eq A x z.
-intro.elim H1.assumption.
+intros.elim H1.assumption.
qed.
theorem f_equal: \forall A,B:Type.\forall f:A\to B.
\forall x,y:A. eq A x y \to eq B (f x) (f y).
-intro.elim H.apply refl_equal.
+intros.elim H.apply refl_equal.
qed.
theorem f_equal2: \forall A,B,C:Type.\forall f:A\to B \to C.
\forall x1,x2:A. \forall y1,y2:B.
eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
-intro.elim H1.elim H.apply refl_equal.
+intros.elim H1.elim H.apply refl_equal.
qed.
inductive nat : Set \def
theorem pred_Sn : \forall n:nat.
(eq nat n (pred (S n))).
-intro.apply refl_equal.
+intros.apply refl_equal.
qed.
theorem injective_S : \forall n,m:nat.
(eq nat (S n) (S m)) \to (eq nat n m).
-intro.(elim (sym_eq ? ? ? (pred_Sn n))).(elim (sym_eq ? ? ? (pred_Sn m))).
+intros.(elim (sym_eq ? ? ? (pred_Sn n))).(elim (sym_eq ? ? ? (pred_Sn m))).
apply f_equal. assumption.
qed.
theorem not_eq_S : \forall n,m:nat.
Not (eq nat n m) \to Not (eq nat (S n) (S m)).
-intro. simplify.intro.
+intros. simplify.intros.
apply H.apply injective_S.assumption.
qed.
| (S p) \Rightarrow True ].
theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-intro.simplify.intro.
+intros.simplify.intros.
cut (not_zero O).exact Hcut.elim (sym_eq ? ? ? H).
exact I.
qed.
theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
-intro.elim n.apply O_S.apply not_eq_S.assumption.
+intros.elim n.apply O_S.apply not_eq_S.assumption.
qed.
-definition plus : nat \to nat \to nat \def
-let rec plus (n,m:nat) \def
- match n:nat with
+let rec plus n m \def
+ match n with
[ O \Rightarrow m
- | (S p) \Rightarrow S (plus p m) ]
-in
-plus.
+ | (S p) \Rightarrow S (plus p m) ].
theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
qed.
theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
qed.
theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
-intro.elim n.simplify.apply plus_n_O.
+intros.elim n.simplify.apply plus_n_O.
simplify.elim (sym_eq ? ? ? H).apply plus_n_Sm.
qed.
theorem assoc_plus:
\forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
qed.
-definition times : nat \to nat \to nat \def
-let rec times (n,m:nat) \def
- match n:nat with
+let rec times n m \def
+ match n with
[ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ]
-in
-times.
+ | (S p) \Rightarrow (plus m (times p m)) ].
theorem times_n_O: \forall n:nat. eq nat O (times n O).
-intro.elim n.simplify.apply refl_equal.simplify.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.assumption.
qed.
theorem times_n_Sm :
\forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
-intro.elim n.simplify.apply refl_equal.
+intros.elim n.simplify.apply refl_equal.
simplify.apply f_equal.elim H.
apply trans_eq ? ? (plus (plus e m) (times e m)).apply sym_eq.
apply assoc_plus.apply trans_eq ? ? (plus (plus m e) (times e m)).
theorem sym_times :
\forall n,m:nat. eq nat (times n m) (times m n).
-intro.elim n.simplify.apply times_n_O.
+intros.elim n.simplify.apply times_n_O.
simplify.elim (sym_eq ? ? ? H).apply times_n_Sm.
qed.
-definition minus : nat \to nat \to nat \def
-let rec minus (n,m:nat) \def
- [\lambda n:nat.nat] match n:nat with
+let rec minus n m \def
+ match n with
[ O \Rightarrow O
| (S p) \Rightarrow
- [\lambda n:nat.nat] match m:nat with
+ match m with
[O \Rightarrow (S p)
- | (S q) \Rightarrow minus p q ]]
-in
-minus.
+ | (S q) \Rightarrow minus p q ]].
theorem nat_case :
\forall n:nat.\forall P:nat \to Prop.
P O \to (\forall m:nat. P (S m)) \to P n.
-intro.elim n.assumption.apply H1.
+intros.elim n.assumption.apply H1.
qed.
theorem nat_double_ind :
(\forall n:nat. R O n) \to
(\forall n:nat. R (S n) O) \to
(\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
-intro.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
-apply nat_case m1.apply H1.intro.apply H2. apply H3.
+intros.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
+apply nat_case m1.apply H1.intros.apply H2. apply H3.
qed.
inductive bool : Set \def
| le_S : \forall m:nat. le n m \to le n (S m).
theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p.
-intro.
+intros.
elim H1.assumption.
apply le_S.assumption.
qed.
theorem le_n_S: \forall n,m:nat. le n m \to le (S n) (S m).
-intro.elim H.
+intros.elim H.
apply le_n.apply le_S.assumption.
qed.
theorem le_O_n : \forall n:nat. le O n.
-intro.elim n.apply le_n.apply le_S. assumption.
+intros.elim n.apply le_n.apply le_S. assumption.
qed.
theorem le_n_Sn : \forall n:nat. le n (S n).
-intro. apply le_S.apply le_n.
+intros. apply le_S.apply le_n.
qed.
theorem le_pred_n : \forall n:nat. le (pred n) n.
-intro.elim n.simplify.apply le_n.simplify.
+intros.elim n.simplify.apply le_n.simplify.
apply le_n_Sn.
qed.
theorem not_zero_le : \forall n,m:nat. (le (S n) m ) \to not_zero m.
-intro.elim H.exact I.exact I.
+intros.elim H.exact I.exact I.
qed.
theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
-intro.simplify.intro.apply not_zero_le ? O H.
+intros.simplify.intros.apply not_zero_le ? O H.
qed.
theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n).
-intro.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
+intros.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
elim n.apply refl_equal.apply False_ind.apply (le_Sn_O ? H2).
qed.
theorem le_S_n : \forall n,m:nat. le (S n) (S m) \to le n m.
-intro.cut le (pred (S n)) (pred (S m)).exact Hcut.
+intros.cut le (pred (S n)) (pred (S m)).exact Hcut.
elim H.apply le_n.apply trans_le ? (pred x).assumption.
apply le_pred_n.
qed.
theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
-intro.elim n.apply le_Sn_O.simplify.intro.
+intros.elim n.apply le_Sn_O.simplify.intros.
cut le (S e) e.apply H.assumption.apply le_S_n.assumption.
qed.
theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).
-intro.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
+intros.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
apply nat_double_ind (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
-intro.whd.intro.
+intros.whd.intros.
apply le_n_O_eq.assumption.
-intro.whd.intro.apply sym_eq.apply le_n_O_eq.assumption.
-intro.whd.intro.apply f_equal.apply H2.
+intros.whd.intros.apply sym_eq.apply le_n_O_eq.assumption.
+intros.whd.intros.apply f_equal.apply H2.
apply le_S_n.assumption.
apply le_S_n.assumption.
qed.
-definition leb : nat \to nat \to bool \def
-let rec leb (n,m:nat) \def
- [\lambda n:nat.bool] match n:nat with
+let rec leb n m \def
+ match n with
[ O \Rightarrow true
| (S p) \Rightarrow
- [\lambda n:nat.bool] match m:nat with
+ match m with
[ O \Rightarrow false
- | (S q) \Rightarrow leb p q]]
-in leb.
+ | (S q) \Rightarrow leb p q]].
theorem le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)).
intros.
simplify.exact le_Sn_O.
intros 2.simplify.elim (leb n1 m1).
simplify.apply le_n_S.apply H.
-simplify.intro.apply H.apply le_S_n.assumption.
+simplify.intros.apply H.apply le_S_n.assumption.
+qed.
+
+inductive Z : Set \def
+ OZ : Z
+| pos : nat \to Z
+| neg : nat \to Z.
+
+definition Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow OZ
+| (S n)\Rightarrow pos n].
+
+coercion Z_of_nat.
+
+definition neg_Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow OZ
+| (S n)\Rightarrow neg n].
+
+definition absZ \def
+\lambda z.
+ match z with
+[ OZ \Rightarrow O
+| (pos n) \Rightarrow n
+| (neg n) \Rightarrow n].
+
+definition OZ_testb \def
+\lambda z.
+match z with
+[ OZ \Rightarrow true
+| (pos n) \Rightarrow false
+| (neg n) \Rightarrow false].
+
+theorem OZ_discr :
+\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)).
+intros.elim z.simplify.apply refl_equal.
+simplify.intros.
+cut match neg e with
+[ OZ \Rightarrow True
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+simplify.intros.
+cut match pos e with
+[ OZ \Rightarrow True
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+qed.
+
+definition Zsucc \def
+\lambda z. match z with
+[ OZ \Rightarrow pos O
+| (pos n) \Rightarrow pos (S n)
+| (neg n) \Rightarrow
+ match n with
+ [ O \Rightarrow OZ
+ | (S p) \Rightarrow neg p]].
+
+definition Zpred \def
+\lambda z. match z with
+[ OZ \Rightarrow neg O
+| (pos n) \Rightarrow
+ match n with
+ [ O \Rightarrow OZ
+ | (S p) \Rightarrow pos p]
+| (neg n) \Rightarrow neg (S n)].
+
+theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+intros.elim z.apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+apply refl_equal.
+qed.
+
+theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+intros.elim z.apply refl_equal.
+apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+qed.
+
+inductive compare :Set \def
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow
+ match m with
+ [ O \Rightarrow EQ
+ | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow
+ match m with
+ [ O \Rightarrow GT
+ | (S q) \Rightarrow nat_compare p q]].
+
+definition compare_invert: compare \to compare \def
+ \lambda c.
+ match c with
+ [ LT \Rightarrow GT
+ | EQ \Rightarrow EQ
+ | GT \Rightarrow LT ].
+
+theorem nat_compare_invert: \forall n,m:nat.
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros.
+apply nat_double_ind (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intro.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intros.simplify.elim H.apply refl_equal.
+qed.
+
+let rec Zplus x y : Z \def
+ match x with
+ [ OZ \Rightarrow y
+ | (pos m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow (pos (S (plus m n)))
+ | (neg n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (neg (pred (minus n m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (pos (pred (minus m n)))]]
+ | (neg m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (pos (pred (minus n m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (neg (pred (minus m n)))]
+ | (neg n) \Rightarrow (neg (S (plus m n)))]].
+
+theorem Zplus_z_O: \forall z:Z. eq Z (Zplus z OZ) z.
+intro.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
+intros.elim x.simplify.elim (sym_eq ? ? ? (Zplus_z_O y)).apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_plus e e1).apply refl_equal.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+simplify.elim (sym_plus e1 e).apply refl_equal.
+qed.
+
+theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+intros.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+intros.elim z.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pp :
+\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_O ?).apply refl_equal.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pn :
+\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_np :
+\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_nn:
+\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+simplify.elim (sym_eq ? ? ? (plus_n_Sm ? ?)).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred:
+\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
+intros.
+elim x. elim y.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim (Zsucc_pos ?).elim (sym_eq ? ? ? (Zsucc_pred ?)).apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus (Zpred OZ) ?.
+elim (Zpred_neg ?).elim (sym_eq ? ? ? (Zpred_succ ?)).
+simplify.apply refl_equal.
+apply Zplus_succ_pred_nn.
+apply Zplus_succ_pred_np.
+elim y.simplify.apply refl_equal.
+apply Zplus_succ_pred_pn.
+apply Zplus_succ_pred_pp.
+qed.
+
+theorem Zsucc_plus_pp :
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zsucc_plus_pn :
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_pn ? m1).
+elim H.apply refl_equal.
qed.
+
+theorem Zsucc_plus_nn :
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_nn ? m1).
+apply refl_equal.
+qed.
+
+theorem Zsucc_plus_np :
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim H.
+elim (Zplus_succ_pred_np ? (S m1)).
+apply refl_equal.
+qed.
+
+
+theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+intros.elim x.elim y.
+simplify. apply refl_equal.
+elim (Zsucc_pos ?).apply refl_equal.
+simplify.apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus OZ ?.simplify.apply refl_equal.
+apply Zsucc_plus_nn.
+apply Zsucc_plus_np.
+elim y.
+elim (sym_Zplus OZ ?).apply refl_equal.
+apply Zsucc_plus_pn.
+apply Zsucc_plus_pp.
+qed.
+
+theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+intros.
+cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
+elim (sym_eq ? ? ? Hcut).
+elim (sym_eq ? ? ? (Zsucc_plus ? ?)).
+elim (sym_eq ? ? ? (Zpred_succ ?)).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zsucc_pred ?)).
+apply refl_equal.
+qed.
+
+theorem assoc_Zplus :
+\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
+intros.elim x.simplify.apply refl_equal.
+elim e.elim (Zpred_neg (Zplus y z)).
+elim (Zpred_neg y).
+elim (Zpred_plus ? ?).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (Zplus (neg e1) y) ?)).
+apply f_equal.assumption.
+elim e.elim (Zsucc_pos ?).
+elim (Zsucc_pos ?).
+apply (sym_eq ? ? ? (Zsucc_plus ? ?)) .
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (Zplus (pos e1) y) ?)).
+apply f_equal.assumption.
+qed.
+
+
+
projects / helm.git / blobdiff