- revision of ground_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / ynat / ynat_plus.ma

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 e8390c8dd2526da1907854a5339e5fb09c9acf88..f7a15c310424605c3f37d7a9a8c86013c590e6b7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
@@ -24,6 +24,15 @@ definition yplus: ynat → ynat → ynat ≝ λx,y. match y with
 
 interpretation "ynat plus" 'plus x y = (yplus x y).
 
+lemma yplus_O2: ∀m:ynat. m + 0 = m.
+// qed.
+
+lemma yplus_S2: ∀m:ynat. ∀n. m + S n = ⫯(m + n).
+// qed.
+
+lemma yplus_Y2: ∀m:ynat. m + (∞) = ∞.
+// qed.
+
 (* Properties on successor **************************************************)
 
 lemma yplus_succ2: ∀m,n. m + ⫯n = ⫯(m + n).
@@ -31,7 +40,7 @@ lemma yplus_succ2: ∀m,n. m + ⫯n = ⫯(m + n).
 qed.
 
 lemma yplus_succ1: ∀m,n. ⫯m + n = ⫯(m + n).
-#m * normalize //
+#m * // #n elim n -n //
 qed.
 
 lemma yplus_succ_swap: ∀m,n. m + ⫯n = ⫯m + n.
@@ -44,18 +53,17 @@ qed.
 (* Basic properties *********************************************************)
 
 lemma yplus_inj: ∀n,m. yinj m + yinj n = yinj (m + n).
-#n elim n -n [ normalize // ]
+#n elim n -n //
 #n #IHn #m >(yplus_succ2 ? n) >IHn -IHn
 <plus_n_Sm //
 qed.
 
 lemma yplus_Y1: ∀m. ∞ + m = ∞.
-* normalize //
+* // #m elim m -m //
 qed.
 
 lemma yplus_comm: commutative … yplus.
 * [ #m ] * [1,3: #n ] //
-normalize >ysucc_iter_Y //
 qed.
 
 lemma yplus_assoc: associative … yplus.
@@ -68,6 +76,10 @@ qed.
 lemma yplus_O1: ∀n:ynat. 0 + n = n.
 #n >yplus_comm // qed.
 
+lemma yplus_comm_23: ∀x,y,z. x + z + y = x + y + z.
+#x #y #z >yplus_assoc //
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
@@ -104,7 +116,7 @@ lemma monotonic_yle_plus_sn: ∀x,y. x ≤ y → ∀z. z + x ≤ z + y.
 
 lemma monotonic_yle_plus: ∀x1,y1. x1 ≤ y1 → ∀x2,y2. x2 ≤ y2 →
                           x1 + x2 ≤ y1 + y2.
-/3 width=3 by monotonic_yle_plus_dx, yle_trans/ qed.
+/3 width=3 by monotonic_yle_plus_dx, monotonic_yle_plus_sn, yle_trans/ qed.
 
 (* Forward lemmas on order **************************************************)
 
@@ -132,6 +144,10 @@ lemma yle_fwd_plus_ge_inj: ∀m1:nat. ∀m2,n1,n2:ynat. m2 ≤ m1 → m1 + n1 
 #x #H0 #H destruct /3 width=4 by yle_fwd_plus_ge, yle_inj/
 qed-.
 
+lemma yle_fwd_plus_yge: ∀n2,m1:ynat. ∀n1,m2:nat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+* // #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.
@@ -155,6 +171,17 @@ 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-.
+
 (* Properties on minus ******************************************************)
 
 lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
@@ -166,10 +193,18 @@ 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.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
+#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-.
@@ -188,12 +223,36 @@ lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z
 ]
 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-.
+
 (* Inversion lemmas on minus ************************************************)
 
 lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.