update in ground_2, static_2, 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 e710032b0391c140715390448a1bd7505b873f4b..2e1d05077a37e9fd25e5d328ba893097629c97e9 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_plus.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_3_2.ma".
 include "ground_2/ynat/ynat_lt.ma".
 
 (* NATURAL NUMBERS WITH INFINITY ********************************************)
@@ -27,7 +28,7 @@ interpretation "ynat plus" 'plus x y = (yplus x y).
 lemma yplus_O2: ∀m:ynat. m + 0 = m.
 // qed.
 
-lemma yplus_S2: â\88\80m:ynat. â\88\80n. m + S n = ⫯(m + n).
+lemma yplus_S2: â\88\80m:ynat. â\88\80n. m + S n = â\86\91(m + n).
 // qed.
 
 lemma yplus_Y2: ∀m:ynat. m + (∞) = ∞.
@@ -35,18 +36,18 @@ lemma yplus_Y2: ∀m:ynat. m + (∞) = ∞.
 
 (* Properties on successor **************************************************)
 
-lemma yplus_succ2: â\88\80m,n. m + ⫯n = ⫯(m + n).
+lemma yplus_succ2: â\88\80m,n. m + â\86\91n = â\86\91(m + n).
 #m * //
 qed.
 
-lemma yplus_succ1: â\88\80m,n. ⫯m + n = ⫯(m + n).
+lemma yplus_succ1: â\88\80m,n. â\86\91m + n = â\86\91(m + n).
 #m * // #n elim n -n //
 qed.
 
-lemma yplus_succ_swap: â\88\80m,n. m + ⫯n = ⫯m + n.
+lemma yplus_succ_swap: â\88\80m,n. m + â\86\91n = â\86\91m + n.
 // qed.
 
-lemma yplus_SO2: â\88\80m. m + 1 = ⫯m.
+lemma yplus_SO2: â\88\80m. m + 1 = â\86\91m.
 * //
 qed.
 
@@ -86,19 +87,19 @@ lemma yplus_comm_24: ∀x1,x2,x3,x4. x1 + x4 + x3 + x2 = x1 + x2 + x3 + x4.
 /2 width=1 by eq_f2/
 qed.
 
-lemma yplus_assoc_23: ∀x1,x2,x3,x4. x1 + x2 + (x3 + x4) = x1 + (x2 + x3) + x4.  
+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.
+lemma yplus_inv_succ_lt_dx: ∀x,y,z:ynat. 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.
+lemma yplus_inv_succ_lt_sn: ∀x,y,z:ynat. 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/
 
@@ -148,15 +149,15 @@ lemma yplus_inv_monotonic_dx_inj: ∀z,x,y. x + yinj z = y + yinj z → x = y.
 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/ 
+#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 
+* /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. 
+                              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-.
@@ -168,13 +169,13 @@ lemma ylt_inv_plus_Y: ∀x,y. x + y < ∞ → x < ∞ ∧ y < ∞.
 #z #H elim (yplus_inv_inj … H) -H /2 width=1 by conj/
 qed-.
 
-lemma ylt_inv_plus_sn: â\88\80x,y. x < y â\86\92 â\88\83â\88\83z. ⫯z + x = y & x < ∞.
+lemma ylt_inv_plus_sn: â\88\80x,y. x < y â\86\92 â\88\83â\88\83z. â\86\91z + 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: â\88\80x,y. x < y â\86\92 â\88\83â\88\83z. x + ⫯z = y & x < ∞.
+lemma ylt_inv_plus_dx: â\88\80x,y. x < y â\86\92 â\88\83â\88\83z. x + â\86\91z = y & x < ∞.
 #x #y #H elim (ylt_inv_plus_sn … H) -H
 #z >yplus_comm /2 width=2 by ex2_intro/
 qed-.
@@ -269,7 +270,7 @@ lemma monotonic_ylt_plus_sn_inj: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z
 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.  
+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.
@@ -302,11 +303,11 @@ qed-.
 
 (* Properties on predeccessor ***********************************************)
 
-lemma yplus_pred1: â\88\80x,y:ynat. 0 < x â\86\92 â«°x + y = â«°(x+y).
+lemma yplus_pred1: â\88\80x,y:ynat. 0 < x â\86\92 â\86\93x + y = â\86\93(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: â\88\80x,y:ynat. 0 < y â\86\92 x + â«°y = â«°(x+y).
+lemma yplus_pred2: â\88\80x,y:ynat. 0 < y â\86\92 x + â\86\93y = â\86\93(x+y).
 /2 width=1 by yplus_pred1/ qed-.