(* *)
(**************************************************************************)
-include "ground_2/notation/functions/successor_1.ma".
-include "ground_2/notation/functions/predecessor_1.ma".
+include "ground_2/notation/functions/uparrow_1.ma".
+include "ground_2/notation/functions/downarrow_1.ma".
include "arithmetics/nat.ma".
-include "ground_2/lib/star.ma".
+include "ground_2/lib/relations.ma".
(* ARITHMETICAL PROPERTIES **************************************************)
-interpretation "nat successor" 'Successor m = (S m).
+interpretation "nat successor" 'UpArrow m = (S m).
-interpretation "nat predecessor" 'Predecessor m = (pred m).
+interpretation "nat predecessor" 'DownArrow m = (pred m).
interpretation "nat min" 'and x y = (min x y).
lemma pred_S: ∀m. pred (S m) = m.
// qed.
+lemma plus_S1: ∀x,y. ↑(x+y) = (↑x) + y.
+// qed.
+
lemma max_O1: ∀n. n = (0 ∨ n).
// qed.
lemma max_O2: ∀n. n = (n ∨ 0).
// qed.
-lemma max_SS: â\88\80n1,n2. ⫯(n1â\88¨n2) = (⫯n1 â\88¨ ⫯n2).
+lemma max_SS: â\88\80n1,n2. â\86\91(n1â\88¨n2) = (â\86\91n1 â\88¨ â\86\91n2).
#n1 #n2 elim (decidable_le n1 n2) #H normalize
[ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
qed.
(* Equations ****************************************************************)
+lemma plus_SO: ∀n. n + 1 = ↑n.
+// qed.
+
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
qed-.
+lemma lt_succ_pred: ∀m,n. n < m → m = ↑↓m.
+#m #n #Hm >S_pred /2 width=2 by ltn_to_ltO/
+qed-.
+
fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
/2 width=1 by plus_minus_minus_be/ qed-.
// qed.
lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
-/2 by plus_minus/ qed.
+/2 by plus_minus/ qed-.
lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
-/2 by plus_minus/ qed.
+/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.
+qed-.
lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1 by arith_b1/
-qed.
+qed-.
lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
#a1 #a2 #b #c1 #H1 #H2 >plus_minus /2 width=1 by arith_b2/
-qed.
+qed-.
lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
/2 width=1 by plus_minus/ qed-.
+lemma plus_to_minus_2: ∀m1,m2,n1,n2. n1 ≤ m1 → n2 ≤ m2 →
+ m1+n2 = m2+n1 → m1-n1 = m2-n2.
+/2 width=1 by arith_b1/ qed-.
+
+lemma idempotent_max: ∀n:nat. n = (n ∨ n).
+#n normalize >le_to_leb_true //
+qed.
+
+lemma associative_max: associative … max.
+#x #y #z normalize
+@(leb_elim x y) normalize #Hxy
+@(leb_elim y z) normalize #Hyz //
+[1,2: >le_to_leb_true /2 width=3 by transitive_le/
+| >not_le_to_leb_false /4 width=3 by lt_to_not_le, not_le_to_lt, transitive_lt/
+ >not_le_to_leb_false //
+]
+qed.
+
(* Properties ***************************************************************)
lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
/3 width=1 by monotonic_le_minus_l/ qed.
(* Note: this might interfere with nat.ma *)
-lemma monotonic_lt_pred: ∀m,n. m < n → O < m → pred m < pred n.
+lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
#m #n #Hmn #Hm whd >(S_pred … Hm)
@le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
qed.
-lemma lt_S_S: â\88\80x,y. x < y â\86\92 ⫯x < ⫯y.
+lemma lt_S_S: â\88\80x,y. x < y â\86\92 â\86\91x < â\86\91y.
/2 width=1 by le_S_S/ qed.
-lemma lt_S: â\88\80n,m. n < m â\86\92 n < ⫯m.
+lemma lt_S: â\88\80n,m. n < m â\86\92 n < â\86\91m.
/2 width=1 by le_S/ qed.
-lemma max_S1_le_S: â\88\80n1,n2,n. (n1 â\88¨ n2) â\89¤ n â\86\92 (⫯n1 â\88¨ n2) â\89¤ ⫯n.
+lemma max_S1_le_S: â\88\80n1,n2,n. (n1 â\88¨ n2) â\89¤ n â\86\92 (â\86\91n1 â\88¨ n2) â\89¤ â\86\91n.
/4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
-lemma max_S2_le_S: â\88\80n1,n2,n. (n1 â\88¨ n2) â\89¤ n â\86\92 (n1 â\88¨ ⫯n2) â\89¤ ⫯n.
+lemma max_S2_le_S: â\88\80n1,n2,n. (n1 â\88¨ n2) â\89¤ n â\86\92 (n1 â\88¨ â\86\91n2) â\89¤ â\86\91n.
/2 width=1 by max_S1_le_S/ qed-.
lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
(* Inversion & forward lemmas ***********************************************)
-lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
-// qed-.
-
-lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
-/2 width=2 by le_plus_minus_comm/ qed-.
-
-lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
-/2 width=1 by monotonic_pred/ qed-.
-
lemma lt_refl_false: ∀n. n < n → ⊥.
#n #H elim (lt_to_not_eq … H) -H /2 width=1 by/
qed-.
lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
-lemma succ_inv_refl_sn: ∀x. ⫯x = x → ⊥.
-#x #H @(lt_le_false x (⫯x)) //
-qed-.
-
-lemma lt_inv_O1: ∀n. 0 < n → ∃m. ⫯m = n.
-* /2 width=2 by ex_intro/
-#H cases (lt_le_false … H) -H //
-qed-.
-
-lemma lt_inv_S1: ∀m,n. ⫯m < n → ∃∃p. m < p & ⫯p = n.
-#m * /3 width=3 by lt_S_S_to_lt, ex2_intro/
-#H cases (lt_le_false … H) -H //
-qed-.
-
-lemma lt_inv_gen: ∀y,x. x < y → ∃∃z. x ≤ z & ⫯z = y.
-* /3 width=3 by le_S_S_to_le, ex2_intro/
-#x #H elim (lt_le_false … H) -H //
-qed-.
-
-lemma pred_inv_refl: ∀m. pred m = m → m = 0.
-* // normalize #m #H elim (lt_refl_false m) //
+lemma succ_inv_refl_sn: ∀x. ↑x = x → ⊥.
+#x #H @(lt_le_false x (↑x)) //
qed-.
lemma le_plus_xSy_O_false: ∀x,y. x + S y ≤ 0 → ⊥.
lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
/2 width=4 by plus_xySz_x_false/ qed-.
+lemma pred_inv_refl: ∀m. pred m = m → m = 0.
+* // normalize #m #H elim (lt_refl_false m) //
+qed-.
+
+lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
+// qed-.
+
+lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
+/2 width=2 by le_plus_minus_comm/ qed-.
+
+lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
+/2 width=1 by monotonic_pred/ qed-.
+
+lemma plus2_inv_le_sn: ∀m1,m2,n1,n2. m1 + n1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
+#m1 #m2 #n1 #n2 #H #Hm
+lapply (monotonic_le_plus_l n1 … Hm) -Hm >H -H
+/2 width=2 by le_plus_to_le/
+qed-.
+
+lemma lt_S_S_to_lt: ∀x,y. ↑x < ↑y → x < y.
+/2 width=1 by le_S_S_to_le/ qed-.
+
(* Note this should go in nat.ma *)
lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
#x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
#H destruct
qed-.
+lemma lt_inv_O1: ∀n. 0 < n → ∃m. ↑m = n.
+* /2 width=2 by ex_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_S1: ∀m,n. ↑m < n → ∃∃p. m < p & ↑p = n.
+#m * /3 width=3 by lt_S_S_to_lt, ex2_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_gen: ∀y,x. x < y → ∃∃z. x ≤ z & ↑z = y.
+* /3 width=3 by le_S_S_to_le, ex2_intro/
+#x #H elim (lt_le_false … H) -H //
+qed-.
+
+lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
+/2 width=1 by plus_le_0/ qed-.
+
+lemma plus_inv_S3_sn: ∀x1,x2,x3. x1+x2 = ↑x3 →
+ ∨∨ ∧∧ x1 = 0 & x2 = ↑x3
+ | ∃∃y1. x1 = ↑y1 & y1 + x2 = x3.
+* /3 width=1 by or_introl, conj/
+#x1 #x2 #x3 <plus_S1 #H destruct
+/3 width=3 by ex2_intro, or_intror/
+qed-.
+
+lemma plus_inv_S3_dx: ∀x2,x1,x3. x1+x2 = ↑x3 →
+ ∨∨ ∧∧ x2 = 0 & x1 = ↑x3
+ | ∃∃y2. x2 = ↑y2 & x1 + y2 = x3.
+* /3 width=1 by or_introl, conj/
+#x2 #x1 #x3 <plus_n_Sm #H destruct
+/3 width=3 by ex2_intro, or_intror/
+qed-.
+
+lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
+/4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
+qed-.
+
lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
* /2 width=1 by conj/ #x #y normalize #H destruct
qed-.
-lemma lt_S_S_to_lt: ∀x,y. ⫯x < ⫯y → x < y.
-/2 width=1 by le_S_S_to_le/ qed-.
+lemma nat_split: ∀x. x = 0 ∨ ∃y. ↑y = x.
+* /3 width=2 by ex_intro, or_introl, or_intror/
+qed-.
lemma lt_elim: ∀R:relation nat.
- (â\88\80n2. R O (⫯n2)) →
- (â\88\80n1,n2. R n1 n2 â\86\92 R (⫯n1) (⫯n2)) →
+ (â\88\80n2. R O (â\86\91n2)) →
+ (â\88\80n1,n2. R n1 n2 â\86\92 R (â\86\91n1) (â\86\91n2)) →
∀n2,n1. n1 < n2 → R n1 n2.
#R #IH1 #IH2 #n2 elim n2 -n2
[ #n1 #H elim (lt_le_false … H) -H //
lemma le_elim: ∀R:relation nat.
(∀n2. R O (n2)) →
- (â\88\80n1,n2. R n1 n2 â\86\92 R (⫯n1) (⫯n2)) →
+ (â\88\80n1,n2. R n1 n2 â\86\92 R (â\86\91n1) (â\86\91n2)) →
∀n1,n2. n1 ≤ n2 → R n1 n2.
#R #IH1 #IH2 #n1 #n2 @(nat_elim2 … n1 n2) -n1 -n2
/4 width=1 by monotonic_pred/ -IH1 -IH2
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