lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
/2 by plus_minus/ qed.
+lemma minus_minus_comm3: ∀n,x,y,z. n-x-y-z = n-y-z-x.
+// qed.
+
lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
qed.
#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/
qed.
+lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
+/2 width=1 by plus_minus/ qed-.
+
+(* Properties ***************************************************************)
+
+lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
+/3 width=1 by monotonic_le_minus_l/ qed.
+
+lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
+/3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
+
+lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-1+n ≤ y-z-1.
+#z #x #y #n #Hzx #Hxny
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-1 ≤ x-z-1+n.
+#z #x #y #n #Hzx #Hyxn
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
(* Inversion & forward lemmas ***********************************************)
axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
#Hxy elim (H Hxy)
qed-.
+lemma pred_inv_refl: ∀m. pred m = m → m = 0.
+* // normalize #m #H elim (lt_refl_false m) //
+qed-.
+
lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥.
#y #z #x elim x -x
[ #H lapply (le_n_O_to_eq … H) -H
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "∞"
+ non associative with precedence 55
+ for @{ 'Infinity }.
+
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( ⫰ term 55 T )"
+ non associative with precedence 55
+ for @{ 'Predecessor $T }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( ⫯ term 55 T )"
+ non associative with precedence 55
+ for @{ 'Successor $T }.
--- /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 "arithmetics/nat.ma".
+include "ground_2/notation/constructors/infinity_0.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the type of natural numbers with infinity *)
+inductive ynat: Type[0] ≝
+| yinj: nat → ynat
+| Y : ynat
+.
+
+coercion yinj.
+
+interpretation "ynat infinity" 'Infinity = Y.
+
+(* Inversion lemmas *********************************************************)
+
+lemma yinj_inj: ∀m,n. yinj m = yinj n → m = n.
+#m #n #H destruct //
+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_2/ynat/ynat_succ.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* order relation *)
+inductive yle: relation ynat ≝
+| yle_inj: ∀m,n. m ≤ n → yle m n
+| yle_Y : ∀m. yle m (∞)
+.
+
+interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact yle_inv_inj_aux: ∀x,y. x ≤ y → ∀m,n. x = yinj m → y = yinj n → m ≤ n.
+#x #y * -x -y
+[ #x #y #Hxy #m #n #Hx #Hy destruct //
+| #x #m #n #_ #Hy destruct
+]
+qed-.
+
+lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
+/2 width=5 by yle_inv_inj_aux/ qed-.
+
+fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
+#m #x * -m -x
+[ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
+| #m #H destruct
+]
+qed-.
+
+lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
+/2 width =3 by yle_inv_O2_aux/ qed-.
+
+fact yle_inv_Y1_aux: ∀x,n. x ≤ n → x = ∞ → n = ∞.
+#x #n * -x -n //
+#x #n #_ #H destruct
+qed-.
+
+lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
+/2 width=3 by yle_inv_Y1_aux/ qed-.
+
+(* Inversion lemmas on successor ********************************************)
+
+fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → ∃∃n. m ≤ n & y = ⫯n.
+#x #y * -x -y
+[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
+ #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
+ #m #Hnm #H destruct
+ @(ex2_intro … m) /2 width=1 by yle_inj/ (**) (* explicit constructor *)
+| #x #y #H destruct
+ @(ex2_intro … (∞)) /2 width=1 by yle_Y/
+]
+qed-.
+
+lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → ∃∃n. m ≤ n & y = ⫯n.
+/2 width=3 by yle_inv_succ1_aux/ qed-.
+
+lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
+#m #n #H elim (yle_inv_succ1 … H) -H
+#x #Hx #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma yle_refl: reflexive … yle.
+* /2 width=1 by le_n, yle_inj/
+qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma yle_pred_sn: ∀m,n. m ≤ n → ⫰m ≤ n.
+#m #n * -m -n /3 width=3 by transitive_le, yle_inj/
+qed.
+
+lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
+/2 width=1 by yle_refl, yle_pred_sn/ qed.
+
+(* Properties on successor **************************************************)
+
+lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
+#m #n * -m -n /3 width=1 by le_S, yle_inj/
+qed.
+
+lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
+/2 width=1 by yle_refl, yle_succ_dx/ qed.
+
+(* Main properties **********************************************************)
+
+theorem yle_trans: Transitive … yle.
+#x #y * -x -y
+[ #x #y #Hxy * //
+ #z #H lapply (yle_inv_inj … H) -H
+ /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
+| #x #z #H lapply ( yle_inv_Y1 … H) //
+]
+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_2/notation/functions/predecessor_1.ma".
+include "ground_2/lib/arith.ma".
+include "ground_2/ynat/ynat.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the predecessor function *)
+definition ypred: ynat → ynat ≝ λm. match m with
+[ yinj m ⇒ pred m
+| Y ⇒ Y
+].
+
+interpretation "ynat predecessor" 'Predecessor m = (ypred m).
+
+(* Properties ***************************************************************)
+
+lemma ypred_inj_rew: ∀m:nat. ⫰m = pred m.
+// qed.
+
+(* Inversion lemmas *********************************************************)
+
+lemma ypred_inv_refl: ∀m. ⫰m = m → m = 0 ∨ m = ∞.
+* // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+/4 width=1 by pred_inv_refl, or_introl, eq_f/
+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_2/notation/functions/successor_1.ma".
+include "ground_2/ynat/ynat_pred.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the successor function *)
+definition ysucc: ynat → ynat ≝ λm. match m with
+[ yinj m ⇒ S m
+| Y ⇒ Y
+].
+
+interpretation "ynat successor" 'Successor m = (ysucc m).
+
+(* Properties ***************************************************************)
+
+lemma ypred_succ: ∀m. ⫰⫯m = m.
+* // qed.
+
+(* Inversion lemmas *********************************************************)
+
+lemma ysucc_inj: ∀m,n. ⫯m = ⫯n → m = n.
+#m #n #H <(ypred_succ m) <(ypred_succ n) //
+qed-.
+
+lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
+* // normalize
+#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+#H elim (lt_refl_false m) //
+qed-.
+
+lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
+ ∃∃m1. n1 = yinj m1 & m2 = S m1.
+#m2 * normalize
+[ #n1 #H destruct /2 width=3 by ex2_intro/
+| #H destruct
+]
+qed-.
+
+lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2 →
+ ∃∃m1. n1 = yinj m1 & m2 = S m1.
+/2 width=1 by ysucc_inv_inj_sn/ qed-.
#m #n elim (decidable_le m n) /2/ /4/
qed-.
+lemma le_inv_S1: ∀x,y. S x ≤ y → ∃∃z. x ≤ z & y = S z.
+#x #y #H elim H -y /2 width=3 by ex2_intro/
+#y #_ * #n #Hxn #H destruct /3 width=3 by le_S, ex2_intro/
+qed-.
+
(* More general conclusion **************************************************)
theorem nat_ind_plus: ∀R:predicate nat.
;;
let predefined_classes = [
- ["!"; "¡"; ];
+ ["!"; "¡"; "⫯"; "⫰"; ];
["?"; "¿"; "⸮"; ];
[":"; "⁝"; ];
["."; "•"; "◦"; ];
["#"; "♯"; "⋕"; "⌘"; ];
["-"; "÷"; "⊢"; "⊩"; ];
- ["="; "≃"; "≈"; "≝"; "≡"; "≅"; "≐"; "≑"; ];
+ ["="; "≃"; "≈"; "≝"; "≡"; "≅"; "≐"; "≑"; "≚"; "≙"; "⌆"; ];
["→"; "↦"; "⇝"; "⤞"; "⇾"; "⤍"; "⤏"; "⤳"; ] ;
["⇒"; "⤇"; "➾"; "⇨"; "➡"; "➤"; "➸"; "⇉"; "⥰"; ] ;
["^"; "↑"; ] ;
["5"; "𝟝"; "⑤"; "⓹"; ] ;
["6"; "𝟞"; "⑥"; "⓺"; ] ;
["7"; "𝟟"; "⑦"; "⓻"; ] ;
- ["8"; "𝟠"; "⑧"; "⓼"; ] ;
+ ["8"; "𝟠"; "⑧"; "⓼"; "∞"; ] ;
["9"; "𝟡"; "⑨"; "⓽"; ] ;
]
;;
projects / helm.git / commitdiff