set "baseuri" "cic:/matita/Z/orders".
include "Z/z.ma".
+include "nat/orders.ma".
definition Zle : Z \to Z \to Prop \def
\lambda x,y:Z.
| (pos n) \Rightarrow
match y with
[ OZ \Rightarrow False
- | (pos m) \Rightarrow (le n m)
+ | (pos m) \Rightarrow n \leq m
| (neg m) \Rightarrow False ]
| (neg n) \Rightarrow
match y with
[ OZ \Rightarrow True
| (pos m) \Rightarrow True
- | (neg m) \Rightarrow (le m n) ]].
+ | (neg m) \Rightarrow m \leq n ]].
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer 'less or equal to'" 'leq x y = (cic:/matita/Z/orders/Zle.con x y).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer 'neither less nor equal to'" 'nleq x y =
+ (cic:/matita/logic/connectives/Not.con (cic:/matita/Z/orders/Zle.con x y)).
definition Zlt : Z \to Z \to Prop \def
\lambda x,y:Z.
| (pos n) \Rightarrow
match y with
[ OZ \Rightarrow False
- | (pos m) \Rightarrow (lt n m)
+ | (pos m) \Rightarrow n<m
| (neg m) \Rightarrow False ]
| (neg n) \Rightarrow
match y with
[ OZ \Rightarrow True
| (pos m) \Rightarrow True
- | (neg m) \Rightarrow (lt m n) ]].
+ | (neg m) \Rightarrow m<n ]].
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer 'less than'" 'lt x y = (cic:/matita/Z/orders/Zlt.con x y).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer 'not less than'" 'nless x y =
+ (cic:/matita/logic/connectives/Not.con (cic:/matita/Z/orders/Zlt.con x y)).
+
theorem irreflexive_Zlt: irreflexive Z Zlt.
-change with \forall x:Z. Zlt x x \to False.
+change with (\forall x:Z. x < x \to False).
intro.elim x.exact H.
-cut (Zlt (neg n) (neg n)) \to False.
-apply Hcut.apply H.simplify.apply not_le_Sn_n.
-cut (Zlt (pos n) (pos n)) \to False.
-apply Hcut.apply H.simplify.apply not_le_Sn_n.
+cut (neg n < neg n \to False).
+apply Hcut.apply H.simplify.unfold lt.apply not_le_Sn_n.
+cut (pos n < pos n \to False).
+apply Hcut.apply H.simplify.unfold lt.apply not_le_Sn_n.
qed.
theorem irrefl_Zlt: irreflexive Z Zlt
\def irreflexive_Zlt.
-definition Z_compare : Z \to Z \to compare \def
-\lambda x,y:Z.
- match x with
- [ OZ \Rightarrow
- match y with
- [ OZ \Rightarrow EQ
- | (pos m) \Rightarrow LT
- | (neg m) \Rightarrow GT ]
- | (pos n) \Rightarrow
- match y with
- [ OZ \Rightarrow GT
- | (pos m) \Rightarrow (nat_compare n m)
- | (neg m) \Rightarrow GT]
- | (neg n) \Rightarrow
- match y with
- [ OZ \Rightarrow LT
- | (pos m) \Rightarrow LT
- | (neg m) \Rightarrow nat_compare m n ]].
-
theorem Zlt_neg_neg_to_lt:
-\forall n,m:nat. Zlt (neg n) (neg m) \to lt m n.
+\forall n,m:nat. neg n < neg m \to m < n.
intros.apply H.
qed.
-theorem lt_to_Zlt_neg_neg: \forall n,m:nat.lt m n \to Zlt (neg n) (neg m).
+theorem lt_to_Zlt_neg_neg: \forall n,m:nat.m < n \to neg n < neg m.
intros.
simplify.apply H.
qed.
theorem Zlt_pos_pos_to_lt:
-\forall n,m:nat. Zlt (pos n) (pos m) \to lt n m.
+\forall n,m:nat. pos n < pos m \to n < m.
intros.apply H.
qed.
-theorem lt_to_Zlt_pos_pos: \forall n,m:nat.lt n m \to Zlt (pos n) (pos m).
+theorem lt_to_Zlt_pos_pos: \forall n,m:nat.n < m \to pos n < pos m.
intros.
simplify.apply H.
qed.
-theorem Z_compare_to_Prop :
-\forall x,y:Z. match (Z_compare x y) with
-[ LT \Rightarrow (Zlt x y)
-| EQ \Rightarrow (eq Z x y)
-| GT \Rightarrow (Zlt y x)].
-intros.
-elim x. elim y.
-simplify.apply refl_eq.
-simplify.exact I.
-simplify.exact I.
-elim y. simplify.exact I.
-simplify.
-cut match (nat_compare n1 n) with
-[ LT \Rightarrow (lt n1 n)
-| EQ \Rightarrow (eq nat n1 n)
-| GT \Rightarrow (lt n n1)] \to
-match (nat_compare n1 n) with
-[ LT \Rightarrow (le (S n1) n)
-| EQ \Rightarrow (eq Z (neg n) (neg n1))
-| GT \Rightarrow (le (S n) n1)].
-apply Hcut. apply nat_compare_to_Prop.
-elim (nat_compare n1 n).
-simplify.exact H.
-simplify.exact H.
-simplify.apply eq_f.apply sym_eq.exact H.
-simplify.exact I.
-elim y.simplify.exact I.
-simplify.exact I.
-simplify.
-cut match (nat_compare n n1) with
-[ LT \Rightarrow (lt n n1)
-| EQ \Rightarrow (eq nat n n1)
-| GT \Rightarrow (lt n1 n)] \to
-match (nat_compare n n1) with
-[ LT \Rightarrow (le (S n) n1)
-| EQ \Rightarrow (eq Z (pos n) (pos n1))
-| GT \Rightarrow (le (S n1) n)].
-apply Hcut. apply nat_compare_to_Prop.
-elim (nat_compare n n1).
-simplify.exact H.
-simplify.exact H.
-simplify.apply eq_f.exact H.
-qed.
-
-theorem Zlt_to_Zle: \forall x,y:Z. Zlt x y \to Zle (Zsucc x) y.
-intros 2.elim x.
-cut (Zlt OZ y) \to (Zle (Zsucc OZ) y).
-apply Hcut. assumption.simplify.elim y.
-simplify.exact H1.
-simplify.exact H1.
-simplify.apply le_O_n.
-cut (Zlt (neg n) y) \to (Zle (Zsucc (neg n)) y).
-apply Hcut. assumption.elim n.
-cut (Zlt (neg O) y) \to (Zle (Zsucc (neg O)) y).
-apply Hcut. assumption.simplify.elim y.
-simplify.exact I.simplify.apply not_le_Sn_O n1 H2.
-simplify.exact I.
-cut (Zlt (neg (S n1)) y) \to (Zle (Zsucc (neg (S n1))) y).
-apply Hcut. assumption.simplify.
-elim y.
-simplify.exact I.
-simplify.apply le_S_S_to_le n2 n1 H3.
-simplify.exact I.
-exact H.
+theorem Zlt_to_Zle: \forall x,y:Z. x < y \to Zsucc x \leq y.
+intros 2.
+elim x.
+(* goal: x=OZ *)
+ cut (OZ < y \to Zsucc OZ \leq y).
+ apply Hcut. assumption.
+ simplify.elim y.
+ simplify.exact H1.
+ simplify.apply le_O_n.
+ simplify.exact H1.
+(* goal: x=pos *)
+ exact H.
+(* goal: x=neg *)
+ cut (neg n < y \to Zsucc (neg n) \leq y).
+ apply Hcut. assumption.
+ elim n.
+ cut (neg O < y \to Zsucc (neg O) \leq y).
+ apply Hcut. assumption.
+ simplify.elim y.
+ simplify.exact I.
+ simplify.exact I.
+ simplify.apply (not_le_Sn_O n1 H2).
+ cut (neg (S n1) < y \to (Zsucc (neg (S n1))) \leq y).
+ apply Hcut. assumption.simplify.
+ elim y.
+ simplify.exact I.
+ simplify.exact I.
+ simplify.apply (le_S_S_to_le n2 n1 H3).
qed.
projects / helm.git / blobdiff