coercion ninj.
-interpretation "zero (non-negative integers" 'Zero = nzero.
+interpretation
+ "zero (non-negative integers)"
+ 'Zero = nzero.
(* Basic inversions *********************************************************)
"cic:/matita/arithmetics/nat/lt_inv_plus_l.con"
"cic:/matita/arithmetics/nat/lt_minus_to_plus.con"
"cic:/matita/arithmetics/nat/lt_minus_to_plus_r.con"
+#
+"cic:/matita/arithmetics/nat/max.con"
+"cic:/matita/arithmetics/nat/commutative_max.con"
####################################
"cic:/matita/arithmetics/nat/to_max.con"
-"cic:/matita/arithmetics/nat/commutative_max.con"
"cic:/matita/arithmetics/nat/leb.con"
"cic:/matita/arithmetics/nat/leb_elim.con"
"cic:/matita/arithmetics/nat/le_maxl.con"
"cic:/matita/arithmetics/nat/le_maxr.con"
"cic:/matita/arithmetics/nat/le_to_leb_true.con"
-"cic:/matita/arithmetics/nat/max.con"
"cic:/matita/arithmetics/nat/not_le_to_leb_false.con"
"cic:/matita/arithmetics/nat/nat_discr.con"
qed.
(*** le_or_ge *)
-lemma nle_ge_e (m) (n): ∨∨ m ≤ n | n ≤ m.
+lemma nle_ge_dis (m) (n): ∨∨ m ≤ n | n ≤ m.
#m @(nat_ind … m) -m [ /2 width=1 by or_introl/ ]
#m #IH #n @(nat_ind … n) -n [ /2 width=1 by or_intror/ ]
#n #_ elim (IH n) -IH /3 width=2 by nle_succ_bi, or_introl, or_intror/
(*** decidable_le *)
lemma nle_dec (m) (n): Decidable … (m ≤ n).
-#m #n elim (nle_ge_e m n) [ /2 width=1 by or_introl/ ]
+#m #n elim (nle_ge_dis m n) [ /2 width=1 by or_introl/ ]
#Hnm elim (eq_nat_dec m n) [ #H destruct /2 width=1 by nle_refl, or_introl/ ]
/4 width=1 by nle_antisym, or_intror/
qed-.
/2 width=1 by nle_succ_bi/ qed.
(*** le_to_or_lt_eq *)
-lemma nle_lt_eq_e (m) (n): m ≤ n → ∨∨ m < n | m = n.
+lemma nle_lt_eq_dis (m) (n): m ≤ n → ∨∨ m < n | m = n.
#m #n * -n /3 width=1 by nle_succ_bi, or_introl/
qed-.
(*** eq_or_gt *)
-lemma eq_gt_e (n): ∨∨ 𝟎 = n | 𝟎 < n.
-#n elim (nle_lt_eq_e (𝟎) n ?)
+lemma eq_gt_dis (n): ∨∨ 𝟎 = n | 𝟎 < n.
+#n elim (nle_lt_eq_dis (𝟎) n ?)
/2 width=1 by or_introl, or_intror/
qed-.
(*** lt_or_ge *)
-lemma nlt_ge_e (m) (n): ∨∨ m < n | n ≤ m.
-#m #n elim (nle_ge_e m n) /2 width=1 by or_intror/
-#H elim (nle_lt_eq_e … H) -H /2 width=1 by nle_refl, or_introl, or_intror/
+lemma nlt_ge_dis (m) (n): ∨∨ m < n | n ≤ m.
+#m #n elim (nle_ge_dis m n) /2 width=1 by or_intror/
+#H elim (nle_lt_eq_dis … H) -H /2 width=1 by nle_refl, or_introl, or_intror/
qed-.
(*** not_le_to_lt *)
lemma le_false_nlt (m) (n): (n ≤ m → ⊥) → m < n.
-#m #n elim (nlt_ge_e m n) [ // ]
+#m #n elim (nlt_ge_dis m n) [ // ]
#H #Hn elim Hn -Hn //
qed.
(* Destructions with nminus *************************************************)
-lemma nlt_fwd_minus_dx (o) (m) (n): m < n - o → o < n.
+lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
#o elim o -o
[ #m #n <nminus_zero_dx
/2 width=3 by le_nlt_trans/
lemma nlt_inv_minus_dx (o) (m) (n): m < n - o → m + o < n.
#o #m #n #Ho
lapply (nle_inv_minus_dx ???? Ho) //
-/3 width=2 by nlt_fwd_minus_dx, nlt_des_le/
+/3 width=2 by nlt_des_minus_dx, nlt_des_le/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground/notation/functions/zero_0.ma".
+include "ground/arith/nat_succ_tri.ma".
+
+(* MAXIMUM FOR NON-NEGATIVE INTEGERS ****************************************)
+
+(*** max *)
+definition nmax: nat → nat → nat ≝
+ λm,n. ntri … m n n n m.
+
+interpretation
+ "maximum (non-negative integers)"
+ 'or m n = (nmax m n).
+
+(* Basic rewrites ***********************************************************)
+
+(*** max_O1 *)
+lemma nmax_zero_sn (n2): n2 = (𝟎 ∨ n2).
+* //
+qed.
+
+(*** max_O2 *)
+lemma nmax_zero_dx (n1): n1 = (n1 ∨ 𝟎).
+* //
+qed.
+
+(*** max_SS *)
+lemma nmax_succ_bi (n1) (n2): ↑(n1 ∨ n2) = (↑n1 ∨ ↑n2).
+#n1 #n2
+@trans_eq [3: @ntri_succ_bi | skip ] (**) (* rewrite fails because δ-expansion gets in the way *)
+<ntri_f_tri //
+qed.
+
+(* Advanced rewrites ********************************************************)
+
+(*** idempotent_max *)
+lemma nmax_idem (n): n = (n ∨ n).
+/2 width=1 by ntri_eq/ qed.
+
+(*** commutative_max *)
+lemma nmax_comm: commutative … nmax.
+#m #n @(nat_ind_2 … m n) -m -n //
+qed-.
+
+(*** associative_max *)
+lemma nmax_assoc: associative … nmax.
+#n1 elim n1 -n1 //
+#n1 #IH #n2 elim n2 -n2 //
+#n2 #_ #n3 elim n3 -n3 //
+qed-.
+
+(* Basic inversions *********************************************************)
+
+(*** max_inv_O3 *)
+lemma nmax_inv_zero (n1) (n2): 𝟎 = (n1 ∨ n2) → ∧∧ 𝟎 = n1 & 𝟎 = n2.
+#n1 elim n1 -n1 [ /2 width=1 by conj/ ]
+#n1 #_ #n2 elim n2 -n2 [ /2 width=1 by conj/ ]
+#n2 #_ <nmax_succ_bi #H
+elim (eq_inv_nzero_succ … H)
+qed-.
λm,n. npred^n m.
interpretation
- "minus (positive integers"
+ "minus (positive integers)"
'minus m n = (nminus m n).
(* Basic rewrites ***********************************************************)
λm,n. nsucc^n m.
interpretation
- "plus (positive integers"
+ "plus (positive integers)"
'plus m n = (nplus m n).
(* Basic rewrites ***********************************************************)
#m #n @(niter_succ … nsucc)
qed.
+(*** plus_S1 *)
lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
#m #n @(niter_appl … nsucc)
qed.
].
interpretation
- "predecessor (non-negative integers"
+ "predecessor (non-negative integers)"
'DownArrow m = (npred m).
(* Basic rewrites ***********************************************************)
+(*** pred_O *)
lemma npred_zero: 𝟎 = ↓𝟎.
// qed.
(* Rewrites with nsucc ******************************************************)
-(*** pred_Sn *)
+(*** pred_Sn pred_S *)
lemma npred_succ (n): n = ↓↑n.
* //
qed.
].
interpretation
- "successor (non-negative integers"
+ "successor (non-negative integers)"
'UpArrow m = (nsucc m).
(* Basic rewrites ***********************************************************)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground/arith/nat_tri.ma".
+include "ground/arith/nat_succ.ma".
+
+(* SUCCESSOR FOR NON-NEGATIVE INTEGERS **************************************)
+
+(* Rewrites with ntri *******************************************************)
+
+lemma ntri_succ_bi (A) (a1) (a2) (a3) (n1) (n2):
+ ntri A (n1) (n2) a1 a2 a3 = ntri A (↑n1) (↑n2) a1 a2 a3.
+#A #a1 #a2 #a3 * [| #p1 ] * //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground/arith/pnat_tri.ma".
+include "ground/arith/nat.ma".
+
+(* TRICHOTOMY OPERATOR FOR NON-NEGATIVE INTEGERS ****************************)
+
+(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
+(*** tri *)
+definition ntri (A:Type[0]) (n1) (n2) (a1) (a2) (a3): A ≝
+ match n1 with
+ [ nzero ⇒ match n2 with [ nzero ⇒ a2 | ninj p2 ⇒ a1 ]
+ | ninj p1 ⇒ match n2 with [ nzero ⇒ a3 | ninj p2 ⇒ ptri A p1 p2 a1 a2 a3 ]
+ ].
+
+(* Basic rewrites ***********************************************************)
+
+lemma ntri_zero_bi (A) (a1) (a2) (a3):
+ a2 = ntri A (𝟎) (𝟎) a1 a2 a3.
+// qed.
+
+lemma ntri_zero_inj (A) (a1) (a2) (a3) (p):
+ a1 = ntri A (𝟎) (ninj p) a1 a2 a3.
+// qed.
+
+lemma ntri_inj_zero (A) (a1) (a2) (a3) (p):
+ a3 = ntri A (ninj p) (𝟎) a1 a2 a3.
+// qed.
+
+lemma ntri_inj_bi (A) (a1) (a2) (a3) (p1) (p2):
+ ptri A (p1) (p2) a1 a2 a3 = ntri A (p1) (p2) a1 a2 a3.
+// qed.
+
+(* Advanced rewrites ********************************************************)
+
+(*** tri_eq *)
+lemma ntri_eq (A) (a1) (a2) (a3) (n): a2 = ntri A n n a1 a2 a3.
+#A #a1 #a2 #a3 * //
+qed.
+
+lemma ntri_f_tri (A) (B) (f) (a1) (a2) (a3) (n1) (n2):
+ f (ntri A n1 n2 a1 a2 a3) = ntri B n1 n2 (f a1) (f a2) (f a3).
+#A #B #f #a1 #a2 #a3
+* [| #p1 ] * // #p2
+<ntri_inj_bi <ntri_inj_bi //
+qed.
λp,q. psucc^q p.
interpretation
- "plus (positive integers"
+ "plus (positive integers)"
'plus p q = (pplus p q).
(* Basic rewrites ***********************************************************)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground/arith/pnat.ma".
+
+(* TRICHOTOMY OPERATOR FOR POSITIVE INTEGERS ********************************)
+
+rec definition ptri (A:Type[0]) p1 p2 a1 a2 a3 on p1 : A ≝
+ match p1 with
+ [ punit ⇒ match p2 with [ punit ⇒ a2 | psucc p2 ⇒ a1 ]
+ | psucc p1 ⇒ match p2 with [ punit ⇒ a3 | psucc p2 ⇒ ptri A p1 p2 a1 a2 a3 ]
+ ].
+
+(* Basic rewrites ***********************************************************)
+
+lemma ptri_unit_bi (A) (a1) (a2) (a3):
+ a2 = ptri A (𝟏) (𝟏) a1 a2 a3.
+// qed.
+
+lemma ptri_unit_succ (A) (a1) (a2) (a3) (p):
+ a1 = ptri A (𝟏) (↑p) a1 a2 a3.
+// qed.
+
+lemma ptri_succ_unit (A) (a1) (a2) (a3) (p):
+ a3 = ptri A (↑p) (𝟏) a1 a2 a3.
+// qed.
+
+lemma ptri_succ_bi (A) (a1) (a2) (a3) (p1) (p2):
+ ptri A (p1) (p2) a1 a2 a3 = ptri A (↑p1) (↑p2) a1 a2 a3.
+// qed.
+
+(* Advanced rewrites ********************************************************)
+
+lemma ptri_eq (A) (a1) (a2) (a3) (p): a2 = ptri A p p a1 a2 a3.
+#A #a1 #a2 #a3 #p elim p -p //
+qed.
+
+lemma ptri_f_tri (A) (B) (f) (a1) (a2) (a3) (p1) (p2):
+ f (ptri A p1 p2 a1 a2 a3) = ptri B p1 p2 (f a1) (f a2) (f a3).
+#A #B #f #a1 #a2 #a3 #p1
+elim p1 -p1 [| #p1 #IH ] * //
+qed.
[ { "non-negative integers" * } {
[ "nat_lt ( ?<? )" "nat_lt_pred" "nat_lt_plus" "nat_lt_minus" "nat_lt_minus_plus" * ]
[ "nat_le ( ?≤? )" "nat_le_pred" "nat_le_plus" "nat_le_minus" "nat_le_minus_plus" * ]
+ [ "nat_max ( ?∨? )" * ]
[ "nat_minus ( ?-? )" "nat_minus_plus" * ]
[ "nat_plus ( ?+? )" * ]
[ "nat_pred ( ↓? )" "nat_pred_succ" * ]
- [ "nat_succ ( ↑? )" "nat_succ_iter" * ]
- [ "nat ( 𝟎 )" "nat_iter ( ?^{?}? )" * ]
+ [ "nat_succ ( ↑? )" "nat_succ_iter" "nat_succ_tri" * ]
+ [ "nat ( 𝟎 )" "nat_iter ( ?^{?}? )" *"nat_tri" ]
}
]
[ { "positive integers" * } {
[ "pnat_plus ( ?+? )" * ]
- [ "pnat ( 𝟏 ) ( ↑? )" "pnat_dis" "pnat_iter ( ?^{?}? )" * ]
+ [ "pnat ( 𝟏 ) ( ↑? )" "pnat_dis" "pnat_iter ( ?^{?}? )" *"pnat_tri" ]
}
]
}
projects / helm.git / commitdiff