- subtraction (and related notions) removed

- more lemmas

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/lib/relations.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/lib/relations.etc
new file mode 100644 (file)
index 0000000..6166cf7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/etc/lib/relations.etc
@@ -0,0 +1,21 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basics/relations.ma".
+
+(* ADDITIONAL PROPERTIES OF RELATIONS ***************************************)
+
+lemma replace2: ∀A,B:Type[0]. ∀R:relation2 A B. ∀x1,y1,x2,y2.
+                R x1 x2 → y1 = x1 → y2 = x2 → R y1 y2.
+// qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_max.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_max.etc
new file mode 100644 (file)
index 0000000..acbd31d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_max.etc
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
+#x #y * -x -y //
+#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
+qed-.
+
+lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
+#x #y * -x -y
+[ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
+| * //
+]
+qed-.
+
+lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
+// qed.
+
+lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
+* // #y * /2 width=1 by yle_inj/
+qed.
+
+lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
+#x #z #Hxz #y #Hyz elim (yle_split x y)
+[ #Hxy >(ymax_pre_dx … Hxy) -x //
+| #Hyx >(ymax_pre_sn … Hyx) -y //
+]
+qed.
+
+lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
+/2 width=1 by ymax_pre_dx/ qed-.
+
+lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
+/2 width=1 by ymax_pre_sn/ qed-.
+
+lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
+// qed.
+
+lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
+/2 width=1 by ymax_pre_i_sn/ qed.
+
+lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
+/2 width=1 by ymax_pre_e/ qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_min.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_min.etc
new file mode 100644 (file)
index 0000000..96b8f2a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_min.etc
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
+#x #y * -x -y
+[ #x #y #Hxy >yminus_inj
+ /3 width=4 by yle_inj, monotonic_le_minus_l/
+| * //
+]
+qed-.
+
+lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
+#x #y * -x -y //
+#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
+qed-.
+
+lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
+#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
+#Hxy >(ymin_pre_sn … Hxy) //
+qed.
+
+lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
+// qed.
+
+lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
+#x #y #H elim (yle_inv_inj2 … H) -H
+#z #Hzx #H destruct >yminus_inj
+/3 width=4 by minus_le_minus_minus_comm, eq_f/
+qed-.
+
+lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
+                  z ≤ yinj x - (yinj x - y).
+#z #x #Hzx #y #Hzy elim (yle_split x y)
+[ #H >(ymin_pre_sn … H) -y //
+| #H >(ymin_pre_dx … H) -x //
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_minus.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_minus.etc
new file mode 100644 (file)
index 0000000..a2f15a8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_minus.etc
@@ -0,0 +1,227 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_lt.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* subtraction *)
+definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
+[ yinj n ⇒ ypred^n x
+| Y      ⇒ yinj 0
+].
+
+interpretation "ynat minus" 'minus x y = (yminus x y).
+
+lemma yminus_O2: ∀m:ynat. m - 0 = m.
+// qed.
+
+lemma yminus_S2: ∀m:ynat. ∀n:nat. m - S n = ⫰(m - n).
+// qed.
+
+lemma yminus_Y2: ∀m. m - (∞) = 0.
+// qed.
+
+(* Basic properties *********************************************************)
+
+lemma yminus_inj: ∀m,n. yinj m - yinj n = yinj (m - n).
+#m #n elim n -n //
+#n #IH >yminus_S2 >IH -IH >eq_minus_S_pred //
+qed.
+
+lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
+#n elim n -n //
+qed.
+
+lemma yminus_O1: ∀x:ynat. 0 - x = 0.
+* // qed.
+
+lemma yminus_refl: ∀x:ynat. x - x = 0.
+* // qed.
+
+lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
+* #y [ * #z [ * // ] ] >yminus_O1 //
+qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma yminus_SO2: ∀m. m - 1 = ⫰m.
+* //
+qed.
+
+lemma yminus_pred1: ∀x,y. ⫰x - y = ⫰(x-y).
+#x * // #y elim y -y //
+qed.
+
+lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
+* // #n *
+[ #m #Hm #Hn >yminus_inj >yminus_inj
+  /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+(* Properties on successor **************************************************)
+
+lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
+* // qed.
+
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
+#n *
+[ #m #Hmn >yminus_inj >yminus_inj
+  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
+* //
+qed.
+
+(* Properties on order ******************************************************)
+
+lemma yle_minus_sn: ∀n,m. m - n ≤ m.
+* // #n * /2 width=1 by yle_inj/
+qed.
+
+lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
+#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
+qed-.
+
+lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
+* // #n *
+[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+  /2 width=1 by yle_inj/
+| >yminus_Y_inj #H destruct
+]
+qed.
+
+lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
+#x #y #Hxy * //
+#z elim z -z /3 width=1 by yle_pred/
+qed.
+
+(* Properties on strict order ***********************************************)
+
+lemma ylt_to_minus: ∀x,y:ynat. x < y → 0 < y - x.
+#x #y #H elim H -x -y /3 width=1 by ylt_inj, lt_plus_to_minus_r/
+qed.
+
+lemma yminus_to_lt: ∀x,y:ynat. 0 < y - x → x < y.
+* [2: #y #H elim (ylt_yle_false … H) // ]
+#m * /4 width=1 by ylt_inv_inj, ylt_inj, lt_minus_to_plus_r/
+qed-.
+
+lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
+#x #y * -x -y
+/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
+qed.
+
+(* Properties on minus ******************************************************)
+
+lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
+#m #n elim n -n //
+#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
+qed.
+
+lemma yplus_minus: ∀m,n. m + n - n ≤ m.
+#m * //
+qed.
+
+lemma yminus_plus2: ∀z,y,x:ynat. x - (y + z) = x - y - z.
+* // #z * [2: >yplus_Y1 >yminus_O1 // ] #y *
+[ #x >yplus_inj >yminus_inj >yminus_inj >yminus_inj /2 width=1 by eq_f/
+| >yplus_inj >yminus_Y_inj //
+]
+qed.
+
+(* Forward lemmas on minus **************************************************)
+
+lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+#x #z #y #H lapply (monotonic_yle_minus_dx … H y) -H //
+qed-.
+
+lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
+
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
+#x *
+[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
+  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yplus_minus_assoc_comm_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z - (y - x) = z + x - y.
+#x *
+[ #y *
+  [ #z >yminus_inj >yminus_inj >yplus_inj >yminus_inj
+    /4 width=1 by yle_inv_inj, minus_le_minus_minus_comm, eq_f/
+  | >yminus_inj >yminus_Y_inj //
+  ]
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
+#y * // #x * //
+#z #Hxy >yplus_inj >yminus_inj <plus_minus
+/2 width=1 by yle_inv_inj/
+qed-.
+
+lemma ylt_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y < z → x < z - y.
+#x #z #y #H lapply (monotonic_ylt_minus_dx … H y ?) -H //
+qed-.
+
+lemma ylt_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x < z → x < z - y.
+/2 width=1 by ylt_plus1_to_minus_inj2/ qed-.
+
+lemma ylt_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. z ≤ x → x < y + z → x - z < y.
+/2 width=1 by monotonic_ylt_minus_dx/ qed-.
+
+lemma ylt_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. z ≤ x → x < z + y → x - z < y.
+/2 width=1 by ylt_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_inv_minus: ∀x1,y1. y1 ≤ yinj x1 →
+                       ∀x2,y2. yinj x1 + x2 = yinj y2 + y1 →
+                       yinj x1 - y1 = yinj y2 - x2 ∧ x2 ≤ yinj y2.
+#x1 #y1 #Hyx1 #x2 #y2 #H0
+lapply (yle_fwd_plus_ge_inj … x2 y2 Hyx1 ?) // #Hxy2
+elim (yle_inv_inj2 … Hyx1) -Hyx1 #m #Hyx1 #H destruct
+elim (yle_inv_inj2 … Hxy2) #n #H1 #H destruct
+>yplus_inj in H0; >yplus_inj >yminus_inj >yminus_inj #H0
+@conj // lapply (yinj_inj … H0) -H0 /3 width=1 by arith_b1, eq_f/
+qed-.
+
+(* Inversion lemmas on minus ************************************************)
+
+lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
+/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
+
+lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
+/2 width=1 by yle_inv_plus_inj2/ qed-.
+
+lemma yle_inv_plus_inj_dx: ∀x,y:ynat. ∀z:nat. x + y ≤ z →
+                           ∃∃m,n. x = yinj m & y = yinj n & x ≤ z - y & y ≤ z.
+#x #y #z #Hz elim (yle_inv_inj2 … Hz)
+#z0 #_ #H elim (yplus_inv_inj … H) -H
+#m #n #H1 #H2 #H3 destruct
+elim (yle_inv_plus_inj2 … Hz) -Hz /2 width=2 by ex4_2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_plus.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_plus.etc
new file mode 100644 (file)
index 0000000..e535b9f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_plus.etc
@@ -0,0 +1,4 @@
+
+lemma pippo: ∀x,y. x + y ≤ x → x = ∞ ∨ y = 0.
+/3 width=1 by discr_yplus_xy_x, yle_antisym/ qed-.
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
index 467d427ff13f7b50e43812e231498a53226ff0bf..0ba823065a3035c73d5492c290385bb296238d88 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
@@ -14,8 +14,7 @@ table {
         [ { "" * } {
            [ "ynat ( ∞ )" "ynat_pred ( ⫰? )" "ynat_succ ( ⫯? )"
              "ynat_le ( ? ≤ ? )" "ynat_lt ( ? < ? )"
-             "ynat_minus ( ? - ? )" "ynat_plus ( ? + ? )"
-             "ynat_max" "ynat_min" *
+             "ynat_plus ( ? + ? )" *
            ]
           }
         ]

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat.ma
index 3dbc0c662b11c6465346474c53e3e2787ce82720..7e2c2dc6c1a1211d4c4920cb02a590a8e2e589e8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "arithmetics/nat.ma".
+include "ground_2/lib/arith.ma".
 include "ground_2/notation/constructors/infinity_0.ma".
 
 (* NATURAL NUMBERS WITH INFINITY ********************************************)
@@ -32,3 +32,11 @@ interpretation "ynat infinity" 'Infinity = Y.
 lemma yinj_inj: ∀m,n. yinj m = yinj n → m = n.
 #m #n #H destruct //
 qed-.
+
+(* Basic properties *********************************************************)
+
+lemma eq_ynat_dec: ∀n1,n2:ynat. Decidable (n1 = n2).
+* [ #n1 ] * [1,3: #n2 ] /2 width=1 by or_introl/
+[2,3: @or_intror #H destruct ]
+elim (eq_nat_dec n1 n2) /4 width=1 by yinj_inj, or_intror, or_introl/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
index b7b57d764ba64147c0a2acf3a34ab21497775f68..613a9793cb6a547b9e53ca49ce0307abaead7230 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
@@ -60,6 +60,11 @@ qed-.
 lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
 /2 width=3 by yle_inv_Y1_aux/ qed-.
 
+lemma yle_antisym: ∀y,x. x ≤ y → y ≤ x → x = y.
+#x #y #H elim H -x -y
+/4 width=1 by yle_inv_Y1, yle_inv_inj, le_to_le_to_eq, eq_f/
+qed-.
+
 (* Basic properties *********************************************************)
 
 lemma le_O1: ∀n:ynat. 0 ≤ n.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
index 3e496e18599bda0201393d76945f5ac93d5538e7..2a9fbb5152e8225158bb3cf212a4e29b429368eb 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
@@ -59,6 +59,11 @@ lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
 #y #H destruct
 qed-.
 
+lemma ylt_inv_Y2: ∀x:ynat. x < ∞ → ∃n. x = yinj n.
+* /2 width=2 by ex_intro/
+#H elim (ylt_inv_Y1 … H)
+qed-.
+
 lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
 * // #n #H lapply (ylt_inv_inj … H) -H normalize
 /3 width=1 by S_pred, eq_f/
@@ -96,7 +101,7 @@ qed-.
 lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
 /2 width=3 by ylt_fwd_succ2_aux/ qed-.
 
-(* inversion and forward lemmas on yle **************************************)
+(* inversion and forward lemmas on order ************************************)
 
 lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
 #m #n * -m -n /2 width=1 by yle_inj/
@@ -119,10 +124,14 @@ lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
 ]
 qed-.
 
+lemma ylt_inv_le: ∀x,y. x < y → x < ∞ ∧ ⫯x ≤ y.
+#x #y #H elim H -x -y /3 width=1 by yle_inj, conj/
+qed-.
+
 (* Basic properties *********************************************************)
 
-lemma ylt_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x.
-* /2 width=1 by/ normalize
+lemma ylt_O1: ∀x. ⫯⫰x = x → 0 < x.
+* // * /2 width=1 by ylt_inj/ normalize
 #H destruct
 qed.
 
@@ -143,6 +152,9 @@ lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
 #m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
 qed.
 
+lemma ylt_succ_Y: ∀x. x < ∞ → ⫯x < ∞.
+* /2 width=1 by/ qed.
+
 lemma yle_succ1_inj: ∀x,y. ⫯yinj x ≤ y → x < y.
 #x * /3 width=1 by yle_inv_inj, ylt_inj/
 qed.
@@ -183,6 +195,15 @@ lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x <
 ]
 qed-.
 
+lemma yle_inv_succ1_lt: ∀x,y. ⫯x ≤ y → 0 < y ∧ x ≤ ⫰y.
+#x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/
+qed-.
+
+lemma yle_lt: ∀x,y. x < ∞ → ⫯x ≤ y → x < y.
+#x * // #y #H elim (ylt_inv_Y2 … H) -H #n #H destruct
+/3 width=1 by ylt_inj, yle_inv_inj/
+qed-.
+
 (* Main properties **********************************************************)
 
 theorem ylt_trans: Transitive … ylt.
@@ -193,3 +214,41 @@ theorem ylt_trans: Transitive … ylt.
 | #x #z #H elim (ylt_yle_false … H) //
 ]
 qed-.
+
+(* Elimination principles ***************************************************)
+
+fact ynat_ind_lt_le_aux: ∀R:predicate ynat.
+                         (∀y. (∀x. x < y → R x) → R y) →
+                         ∀y:nat. ∀x. x ≤ y → R x.
+#R #IH #y elim y -y
+[ #x #H >(yle_inv_O2 … H) -x
+  @IH -IH #x #H elim (ylt_yle_false … H) -H //
+| /5 width=3 by ylt_yle_trans, ylt_fwd_succ2/
+]
+qed-.
+
+fact ynat_ind_lt_aux: ∀R:predicate ynat.
+                      (∀y. (∀x. x < y → R x) → R y) →
+                      ∀y:nat. R y.
+/4 width=2 by ynat_ind_lt_le_aux/ qed-.
+
+lemma ynat_ind_lt: ∀R:predicate ynat.
+                   (∀y. (∀x. x < y → R x) → R y) →
+                   ∀y. R y.
+#R #IH * /4 width=1 by ynat_ind_lt_aux/
+@IH #x #H elim (ylt_inv_Y2 … H) -H
+#n #H destruct /4 width=1 by ynat_ind_lt_aux/
+qed-.
+
+fact ynat_f_ind_aux: ∀A. ∀f:A→ynat. ∀R:predicate A.
+                     (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) →
+                     ∀x,a. f a = x → R a.
+#A #f #R #IH #x @(ynat_ind_lt … x) -x
+/3 width=3 by/
+qed-.
+
+lemma ynat_f_ind: ∀A. ∀f:A→ynat. ∀R:predicate A.
+                  (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) → ∀a. R a.
+#A #f #R #IH #a
+@(ynat_f_ind_aux … IH) -IH [2: // | skip ]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma
deleted file mode 100644 (file)
index acbd31d..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma
+++ /dev/null
@@ -1,58 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
-#x #y * -x -y //
-#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
-qed-.
-
-lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
-#x #y * -x -y
-[ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
-| * //
-]
-qed-.
-
-lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
-// qed.
-
-lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
-* // #y * /2 width=1 by yle_inj/
-qed.
-
-lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
-#x #z #Hxz #y #Hyz elim (yle_split x y)
-[ #Hxy >(ymax_pre_dx … Hxy) -x //
-| #Hyx >(ymax_pre_sn … Hyx) -y //
-]
-qed.
-
-lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
-/2 width=1 by ymax_pre_dx/ qed-.
-
-lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
-/2 width=1 by ymax_pre_sn/ qed-.
-
-lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
-// qed.
-
-lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
-/2 width=1 by ymax_pre_i_sn/ qed.
-
-lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
-/2 width=1 by ymax_pre_e/ qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_min.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_min.ma
deleted file mode 100644 (file)
index 96b8f2a..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_min.ma
+++ /dev/null
@@ -1,52 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
-#x #y * -x -y
-[ #x #y #Hxy >yminus_inj
- /3 width=4 by yle_inj, monotonic_le_minus_l/
-| * //
-]
-qed-.
-
-lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
-#x #y * -x -y //
-#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
-qed-.
-
-lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
-#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
-#Hxy >(ymin_pre_sn … Hxy) //
-qed.
-
-lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
-// qed.
-
-lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
-#x #y #H elim (yle_inv_inj2 … H) -H
-#z #Hzx #H destruct >yminus_inj
-/3 width=4 by minus_le_minus_minus_comm, eq_f/
-qed-.
-
-lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
-                  z ≤ yinj x - (yinj x - y).
-#z #x #Hzx #y #Hzy elim (yle_split x y)
-[ #H >(ymin_pre_sn … H) -y //
-| #H >(ymin_pre_dx … H) -x //
-]
-qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma
deleted file mode 100644 (file)
index 0c80f83..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma
+++ /dev/null
@@ -1,129 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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_lt.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* subtraction *)
-definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
-[ yinj n ⇒ ypred^n x
-| Y      ⇒ yinj 0
-].
-
-interpretation "ynat minus" 'minus x y = (yminus x y).
-
-lemma yminus_O2: ∀m:ynat. m - 0 = m.
-// qed.
-
-lemma yminus_S2: ∀m:ynat. ∀n:nat. m - S n = ⫰(m - n).
-// qed.
-
-lemma yminus_Y2: ∀m. m - (∞) = 0.
-// qed.
-
-(* Basic properties *********************************************************)
-
-lemma yminus_inj: ∀m,n. yinj m - yinj n = yinj (m - n).
-#m #n elim n -n //
-#n #IH >yminus_S2 >IH -IH >eq_minus_S_pred //
-qed.
-
-lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
-#n elim n -n //
-qed.
-
-lemma yminus_O1: ∀x:ynat. 0 - x = 0.
-* // qed.
-
-lemma yminus_refl: ∀x:ynat. x - x = 0.
-* // qed.
-
-lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
-* #y [ * #z [ * // ] ] >yminus_O1 //
-qed.
-
-(* Properties on predecessor ************************************************)
-
-lemma yminus_SO2: ∀m. m - 1 = ⫰m.
-* //
-qed.
-
-lemma yminus_pred1: ∀x,y. ⫰x - y = ⫰(x-y).
-#x * // #y elim y -y //
-qed.
-
-lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
-* // #n *
-[ #m #Hm #Hn >yminus_inj >yminus_inj
-  /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-(* Properties on successor **************************************************)
-
-lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
-* // qed.
-
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
-#n *
-[ #m #Hmn >yminus_inj >yminus_inj
-  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
-* //
-qed.
-
-(* Properties on order ******************************************************)
-
-lemma yle_minus_sn: ∀n,m. m - n ≤ m.
-* // #n * /2 width=1 by yle_inj/
-qed.
-
-lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
-#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
-qed-.
-
-lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
-* // #n *
-[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
-  /2 width=1 by yle_inj/
-| >yminus_Y_inj #H destruct
-]
-qed.
-
-lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
-#x #y #Hxy * //
-#z elim z -z /3 width=1 by yle_pred/
-qed.
-
-(* Properties on strict order ***********************************************)
-
-lemma ylt_to_minus: ∀x,y:ynat. x < y → 0 < y - x.
-#x #y #H elim H -x -y /3 width=1 by ylt_inj, lt_plus_to_minus_r/
-qed.
-
-lemma yminus_to_lt: ∀x,y:ynat. 0 < y - x → x < y.
-* [2: #y #H elim (ylt_yle_false … H) // ]
-#m * /4 width=1 by ylt_inv_inj, ylt_inj, lt_minus_to_plus_r/
-qed-.
-
-lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
-#x #y * -x -y
-/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
-qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
index f7a15c310424605c3f37d7a9a8c86013c590e6b7..e710032b0391c140715390448a1bd7505b873f4b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/ynat/ynat_minus.ma".
+include "ground_2/ynat/ynat_lt.ma".
 
 (* NATURAL NUMBERS WITH INFINITY ********************************************)
 
@@ -80,6 +80,43 @@ lemma yplus_comm_23: ∀x,y,z. x + z + y = x + y + z.
 #x #y #z >yplus_assoc //
 qed.
 
+lemma yplus_comm_24: ∀x1,x2,x3,x4. x1 + x4 + x3 + x2 = x1 + x2 + x3 + x4.
+#x1 #x2 #x3 #x4
+>yplus_assoc >yplus_assoc >yplus_assoc >yplus_assoc
+/2 width=1 by eq_f2/
+qed.
+
+lemma yplus_assoc_23: ∀x1,x2,x3,x4. x1 + x2 + (x3 + x4) = x1 + (x2 + x3) + x4.  
+#x1 #x2 #x3 #x4 >yplus_assoc >yplus_assoc
+/2 width=1 by eq_f2/
+qed.
+
+(* Inversion lemmas on successor *********************************************)
+
+lemma yplus_inv_succ_lt_dx: ∀x,y,z. 0 < y → x + y = ⫯z → x + ⫰y = z.
+#x #y #z #H <(ylt_inv_O1 y) // >yplus_succ2
+/2 width=1 by ysucc_inv_inj/
+qed-.
+
+lemma yplus_inv_succ_lt_sn: ∀x,y,z. 0 < x → x + y = ⫯z → ⫰x + y = z.
+#x #y #z #H <(ylt_inv_O1 x) // >yplus_succ1
+/2 width=1 by ysucc_inv_inj/
+
+qed-.
+
+(* Inversion lemmas on order ************************************************)
+
+lemma yle_inv_plus_dx: ∀x,y. x ≤ y → ∃z. x + z = y.
+#x #y #H elim H -x -y /2 width=2 by ex_intro/
+#m #n #H @(ex_intro … (yinj (n-m))) (**) (* explicit constructor *)
+/3 width=1 by plus_minus, eq_f/
+qed-.
+
+lemma yle_inv_plus_sn: ∀x,y. x ≤ y → ∃z. z + x = y.
+#x #y #H elim (yle_inv_plus_dx … H) -H
+/2 width=2 by ex_intro/
+qed-.
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
@@ -89,6 +126,59 @@ lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
 /3 width=5 by yinj_inj, ex3_2_intro/
 qed-.
 
+lemma yplus_inv_O: ∀x,y:ynat. x + y = 0 → x = 0 ∧ y = 0.
+#x #y #H elim (yplus_inv_inj … H) -H
+#m * /2 width=1 by conj/ #n <plus_n_Sm #H destruct
+qed-.
+
+lemma discr_yplus_xy_x: ∀x,y. x + y = x → x = ∞ ∨ y = 0.
+* /2 width=1 by or_introl/
+#x elim x -x /2 width=1 by or_intror/
+#x #IHx * [2: >yplus_Y2 #H destruct ]
+#y <ysucc_inj >yplus_succ1 #H
+lapply (ysucc_inv_inj … H) -H #H
+elim (IHx … H) -IHx -H /2 width=1 by or_introl, or_intror/
+qed-.
+
+lemma discr_yplus_x_xy: ∀x,y. x = x + y → x = ∞ ∨ y = 0.
+/2 width=1 by discr_yplus_xy_x/ qed-.
+
+lemma yplus_inv_monotonic_dx_inj: ∀z,x,y. x + yinj z = y + yinj z → x = y.
+#z @(nat_ind_plus … z) -z /3 width=2 by ysucc_inv_inj/
+qed-.
+
+lemma yplus_inv_monotonic_dx: ∀z,x,y. z < ∞ → x + z = y + z → x = y.
+#z #x #y #H elim (ylt_inv_Y2 … H) -H /2 width=2 by yplus_inv_monotonic_dx_inj/ 
+qed-.
+
+lemma yplus_inv_Y2: ∀x,y. x + y = ∞ → x = ∞ ∨ y = ∞.
+* /2 width=1 by or_introl/ #x * // #y >yplus_inj #H destruct 
+qed-.
+
+lemma yplus_inv_monotonic_23: ∀z,x1,x2,y1,y2. z < ∞ →
+                              x1 + z + y1 = x2 + z + y2 → x1 + y1 = x2 + y2. 
+#z #x1 #x2 #y1 #y2 #Hz #H @(yplus_inv_monotonic_dx z) //
+>yplus_comm_23 >H -H //
+qed-.
+
+(* Inversion lemmas on strict_order *****************************************)
+
+lemma ylt_inv_plus_Y: ∀x,y. x + y < ∞ → x < ∞ ∧ y < ∞.
+#x #y #H elim (ylt_inv_Y2 … H) -H
+#z #H elim (yplus_inv_inj … H) -H /2 width=1 by conj/
+qed-.
+
+lemma ylt_inv_plus_sn: ∀x,y. x < y → ∃∃z. ⫯z + x = y & x < ∞.
+#x #y #H elim (ylt_inv_le … H) -H
+#Hx #H elim (yle_inv_plus_sn … H) -H
+/2 width=2 by ex2_intro/
+qed-.
+
+lemma ylt_inv_plus_dx: ∀x,y. x < y → ∃∃z. x + ⫯z = y & x < ∞.
+#x #y #H elim (ylt_inv_plus_sn … H) -H
+#z >yplus_comm /2 width=2 by ex2_intro/
+qed-.
+
 (* Properties on order ******************************************************)
 
 lemma yle_plus_dx2: ∀n,m. n ≤ m + n.
@@ -118,6 +208,11 @@ lemma monotonic_yle_plus: ∀x1,y1. x1 ≤ y1 → ∀x2,y2. x2 ≤ y2 →
                           x1 + x2 ≤ y1 + y2.
 /3 width=3 by monotonic_yle_plus_dx, monotonic_yle_plus_sn, yle_trans/ qed.
 
+lemma ylt_plus_Y: ∀x,y. x < ∞ → y < ∞ → x + y < ∞.
+#x #y #Hx elim (ylt_inv_Y2 … Hx) -Hx
+#m #Hm #Hy elim (ylt_inv_Y2 … Hy) -Hy //
+qed.
+
 (* Forward lemmas on order **************************************************)
 
 lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
@@ -126,17 +221,25 @@ lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
 lemma yle_fwd_plus_sn1: ∀x,y,z. x + y ≤ z → x ≤ z.
 /2 width=3 by yle_trans/ qed-.
 
-lemma yle_inv_monotonic_plus_dx: ∀x,y:ynat.∀z:nat. x + z ≤ y + z → x ≤ y.
+lemma yle_inv_monotonic_plus_dx_inj: ∀x,y:ynat.∀z:nat. x + z ≤ y + z → x ≤ y.
 #x #y #z elim z -z /3 width=1 by yle_inv_succ/
 qed-.
 
-lemma yle_inv_monotonic_plus_sn: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
-/2 width=2 by yle_inv_monotonic_plus_dx/ qed-.
+lemma yle_inv_monotonic_plus_sn_inj: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
+/2 width=2 by yle_inv_monotonic_plus_dx_inj/ qed-.
+
+lemma yle_inv_monotonic_plus_dx: ∀x,y,z. z < ∞ → x + z ≤ y + z → x ≤ y.
+#x #y #z #Hz elim (ylt_inv_Y2 … Hz) -Hz #m #H destruct
+/2 width=2 by yle_inv_monotonic_plus_sn_inj/
+qed-.
+
+lemma yle_inv_monotonic_plus_sn: ∀x,y,z. z < ∞ → z + x ≤ z + y → x ≤ y.
+/2 width=3 by yle_inv_monotonic_plus_dx/ qed-.
 
 lemma yle_fwd_plus_ge: ∀m1,m2:nat. m2 ≤ m1 → ∀n1,n2:ynat. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
 #m1 #m2 #Hm12 #n1 #n2 #H
 lapply (monotonic_yle_plus … Hm12 … H) -Hm12 -H
-/2 width=2 by yle_inv_monotonic_plus_sn/
+/2 width=2 by yle_inv_monotonic_plus_sn_inj/
 qed-.
 
 lemma yle_fwd_plus_ge_inj: ∀m1:nat. ∀m2,n1,n2:ynat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
@@ -148,115 +251,62 @@ lemma yle_fwd_plus_yge: ∀n2,m1:ynat. ∀n1,m2:nat. m2 ≤ m1 → m1 + n1 ≤ m
 * // #n2 * /2 width=4 by yle_fwd_plus_ge_inj/
 qed-.
 
-(* Forward lemmas on strict order *******************************************)
-
-lemma ylt_inv_monotonic_plus_dx: ∀x,y,z. x + z < y + z → x < y.
-* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 … H) ]
-#x * // #y * [2: #H elim (ylt_inv_Y1 … H) ]
-/4 width=3 by ylt_inv_inj, ylt_inj, lt_plus_to_lt_l/
-qed-.
-
 (* Properties on strict order ***********************************************)
 
-lemma ylt_plus_dx1_trans: ∀x,z. z < x → ∀y. z < x + yinj y.
+lemma ylt_plus_dx1_trans: ∀x,z. z < x → ∀y. z < x + y.
 /2 width=3 by ylt_yle_trans/ qed.
 
-lemma ylt_plus_dx2_trans: ∀y,z. z < y → ∀x. z < yinj x + y.
+lemma ylt_plus_dx2_trans: ∀y,z. z < y → ∀x. z < x + y.
 /2 width=3 by ylt_yle_trans/ qed.
 
-lemma monotonic_ylt_plus_dx: ∀x,y. x < y → ∀z:nat. x + yinj z < y + yinj z.
+lemma monotonic_ylt_plus_dx_inj: ∀x,y. x < y → ∀z:nat. x + yinj z < y + yinj z.
 #x #y #Hxy #z elim z -z /3 width=1 by ylt_succ/
 qed.
 
-lemma monotonic_ylt_plus_sn: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
-/2 width=1 by monotonic_ylt_plus_dx/ qed.
-
-(* Properties on predeccessor ***********************************************)
-
-lemma yplus_pred1: ∀x,y:ynat. 0 < x → ⫰x + y = ⫰(x+y).
-#x * // #y elim y -y // #y #IH #Hx
->yplus_S2 >yplus_S2 >IH -IH // >ylt_inv_O1
-/2 width=1 by ylt_plus_dx1_trans/
-qed-.
-
-lemma yplus_pred2: ∀x,y:ynat. 0 < y → x + ⫰y = ⫰(x+y).
-/2 width=1 by yplus_pred1/ qed-.
+lemma monotonic_ylt_plus_sn_inj: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
+/2 width=1 by monotonic_ylt_plus_dx_inj/ qed.
 
-(* Properties on minus ******************************************************)
+lemma monotonic_ylt_plus_dx: ∀x,y. x < y → ∀z. z < ∞ → x + z < y + z.
+#x #y #Hxy #z #Hz elim (ylt_inv_Y2 … Hz) -Hz
+#m #H destruct /2 width=1 by monotonic_ylt_plus_dx_inj/
+qed.  
 
-lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
-#m #n elim n -n //
-#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
-qed.
+lemma monotonic_ylt_plus_sn: ∀x,y. x < y → ∀z. z < ∞ → z + x < z + y.
+/2 width=1 by monotonic_ylt_plus_dx/ qed.
 
-lemma yplus_minus: ∀m,n. m + n - n ≤ m.
-#m * //
+lemma monotonic_ylt_plus_inj: ∀m1,m2. m1 < m2 → ∀n1,n2. yinj n1 ≤ n2 → m1 + n1 < m2 + n2.
+/3 width=3 by monotonic_ylt_plus_sn_inj, monotonic_yle_plus_sn, ylt_yle_trans/
 qed.
 
-lemma yminus_plus2: ∀z,y,x:ynat. x - (y + z) = x - y - z.
-* // #z * [2: >yplus_Y1 >yminus_O1 // ] #y *
-[ #x >yplus_inj >yminus_inj >yminus_inj >yminus_inj /2 width=1 by eq_f/
-| >yplus_inj >yminus_Y_inj //
-]
+lemma monotonic_ylt_plus: ∀m1,m2. m1 < m2 → ∀n1. n1 < ∞ → ∀n2. n1 ≤ n2 → m1 + n1 < m2 + n2.
+#m1 #m2 #Hm12 #n1 #H elim (ylt_inv_Y2 … H) -H #m #H destruct /2 width=1 by monotonic_ylt_plus_inj/
 qed.
 
-(* Forward lemmas on minus **************************************************)
-
-lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
-#x #z #y #H lapply (monotonic_yle_minus_dx … H y) -H //
-qed-.
-
-lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
-/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
-
-lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
-
-lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
-/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+(* Forward lemmas on strict order *******************************************)
 
-lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
-#x *
-[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
-  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
+lemma ylt_inv_monotonic_plus_dx: ∀x,y,z. x + z < y + z → x < y.
+* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 … H) ]
+#x * // #y * [2: #H elim (ylt_inv_Y1 … H) ]
+/4 width=3 by ylt_inv_inj, ylt_inj, lt_plus_to_lt_l/
 qed-.
 
-lemma yplus_minus_assoc_comm_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z - (y - x) = z + x - y.
-#x *
-[ #y *
-  [ #z >yminus_inj >yminus_inj >yplus_inj >yminus_inj
-    /4 width=1 by yle_inv_inj, minus_le_minus_minus_comm, eq_f/
-  | >yminus_inj >yminus_Y_inj //
-  ]
-| >yminus_Y_inj //
-]
+lemma ylt_fwd_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 < m2 + n2 → n1 < n2.
+#m1 #m2 #Hm12 #n1 #n2 #H elim (ylt_fwd_gen … H)
+#x #H0 elim (yplus_inv_inj … H0) -H0
+#y #z #_ #H2 #H3 destruct -x
+elim (yle_inv_inj2 … Hm12)
+#x #_ #H0 destruct
+lapply (monotonic_ylt_plus … H … Hm12) -H -Hm12
+/2 width=2 by ylt_inv_monotonic_plus_dx/
 qed-.
 
-lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
-#y * // #x * //
-#z #Hxy >yplus_inj >yminus_inj <plus_minus
-/2 width=1 by yle_inv_inj/
-qed-.
+(* Properties on predeccessor ***********************************************)
 
-lemma ylt_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y < z → x < z - y.
-#x #z #y #H lapply (monotonic_ylt_minus_dx … H y ?) -H //
+lemma yplus_pred1: ∀x,y:ynat. 0 < x → ⫰x + y = ⫰(x+y).
+#x * // #y elim y -y // #y #IH #Hx
+>yplus_S2 >yplus_S2 >IH -IH // >ylt_inv_O1
+/2 width=1 by ylt_plus_dx1_trans/
 qed-.
 
-lemma ylt_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x < z → x < z - y.
-/2 width=1 by ylt_plus1_to_minus_inj2/ qed-.
-
-lemma ylt_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. z ≤ x → x < y + z → x - z < y.
-/2 width=1 by monotonic_ylt_minus_dx/ qed-.
-
-lemma ylt_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. z ≤ x → x < z + y → x - z < y.
-/2 width=1 by ylt_plus2_to_minus_inj2/ qed-.
-
-(* Inversion lemmas on minus ************************************************)
-
-lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
-/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
-
-lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
-/2 width=1 by yle_inv_plus_inj2/ qed-.
+lemma yplus_pred2: ∀x,y:ynat. 0 < y → x + ⫰y = ⫰(x+y).
+/2 width=1 by yplus_pred1/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
index c95c72d152872c8b67f8b063c8be137a911b1c74..c4c989f77d6e2de9869f910d5e67e80682a702ad 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
@@ -88,3 +88,11 @@ qed-.
 
 lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
 /2 width=2 by ysucc_inv_O_sn/ qed-.
+
+(* Eliminators **************************************************************)
+
+lemma ynat_ind: ∀R:predicate ynat.
+                R 0 → (∀n:nat. R n → R (⫯n)) → R (∞) →
+                ∀x. R x.
+#R #H1 #H2 #H3 * // #n elim n -n /2 width=1 by/
+qed-.