(* *)
(**************************************************************************)
+include "arithmetics/nat.ma".
+include "ground_2/xoa/ex_3_1.ma".
+include "ground_2/xoa/or_3.ma".
include "ground_2/notation/functions/uparrow_1.ma".
include "ground_2/notation/functions/downarrow_1.ma".
-include "arithmetics/nat.ma".
+include "ground_2/pull/pull_2.ma".
include "ground_2/lib/relations.ma".
(* ARITHMETICAL PROPERTIES **************************************************)
[ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
qed.
-(* Equations ****************************************************************)
+(* Equalities ***************************************************************)
+
+lemma plus_SO_sn (n): 1 + n = ↑n.
+// qed-.
-lemma plus_SO: ∀n. n + 1 = ↑n.
+lemma plus_SO_dx (n): n + 1 = ↑n.
// qed.
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
-lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
-// qed.
+lemma plus_minus_m_m_commutative (n) (m): m ≤ n → n = m+(n-m).
+/2 width=1 by plus_minus_associative/ qed-.
-lemma arith_l: ∀x. 1 = 1-x+(x-(x-1)).
-* // #x >minus_S_S >minus_S_S <minus_O_n <minus_n_O //
-qed.
+lemma plus_to_minus_2: ∀m1,m2,n1,n2. n1 ≤ m1 → n2 ≤ m2 →
+ m1+n2 = m2+n1 → m1-n1 = m2-n2.
+#m1 #m2 #n1 #n2 #H1 #H2 #H
+@plus_to_minus >plus_minus_associative //
+qed-.
(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
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-.
-
-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-.
-
-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-.
-
-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.
#m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
qed-.
-fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
-// qed-.
-
-fact le_repl_sn_trans_aux: ∀x,y,z:nat. x ≤ z → y = x → y ≤ z.
-// qed-.
-
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 minus_le_trans_sn: ∀x1,x2. x1 ≤ x2 → ∀x. x1-x ≤ x2.
/2 width=3 by transitive_le/ qed.
+lemma le_plus_to_minus_l: ∀a,b,c. a + b ≤ c → b ≤ c-a.
+/2 width=1 by le_plus_to_minus_r/
+qed-.
+
+lemma le_plus_to_minus_comm: ∀n,m,p. n ≤ p+m → n-p ≤ m.
+/2 width=1 by le_plus_to_minus/ qed-.
+
+lemma le_inv_S1: ∀m,n. ↑m ≤ n → ∃∃p. m ≤ p & ↑p = n.
+#m *
+[ #H lapply (le_n_O_to_eq … H) -H
+ #H destruct
+| /3 width=3 by monotonic_pred, ex2_intro/
+]
+qed-.
+
(* Note: this might interfere with nat.ma *)
lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
#m #n #Hmn #Hm whd >(S_pred … Hm)
lemma lt_S: ∀n,m. n < m → n < ↑m.
/2 width=1 by le_S/ qed.
+lemma monotonic_lt_minus_r:
+∀p,q,n. q < n -> q < p → n-p < n-q.
+#p #q #n #Hn #H
+lapply (monotonic_le_minus_r … n H) -H #H
+@(le_to_lt_to_lt … H) -H
+/2 width=1 by lt_plus_to_minus/
+qed.
+
lemma max_S1_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (↑n1 ∨ n2) ≤ ↑n.
/4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
lemma max_S2_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (n1 ∨ ↑n2) ≤ ↑n.
/2 width=1 by max_S1_le_S/ 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 ***********************************************)
lemma lt_refl_false: ∀n. n < n → ⊥.
lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
+lemma le_dec (n) (m): Decidable (n≤m).
+#n elim n -n [ /2 width=1 by or_introl/ ]
+#n #IH * [ /3 width=2 by lt_zero_false, or_intror/ ]
+#m elim (IH m) -IH
+[ /3 width=1 by or_introl, le_S_S/
+| /4 width=1 by or_intror, le_S_S_to_le/
+]
+qed-.
+
lemma succ_inv_refl_sn: ∀x. ↑x = x → ⊥.
#x #H @(lt_le_false x (↑x)) //
qed-.
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) //
+lemma pred_inv_fix_sn: ∀x. ↓x = x → 0 = x.
+* // #x <pred_Sn #H
+elim (succ_inv_refl_sn x) //
qed-.
lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
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
#n1 #H elim (lt_le_false … H) -H //
qed-.
+lemma nat_elim_le_sn (Q:relation …):
+ (∀m1,m2. (∀m. m < m2-m1 → Q (m2-m) m2) → m1 ≤ m2 → Q m1 m2) →
+ ∀n1,n2. n1 ≤ n2 → Q n1 n2.
+#Q #IH #n1 #n2 #Hn
+<(minus_minus_m_m … Hn) -Hn
+lapply (minus_le n2 n1)
+let d ≝ (n2-n1)
+@(nat_elim1 … d) -d -n1 #d
+@pull_2 #Hd
+<(minus_minus_m_m … Hd) in ⊢ (%→?); -Hd
+let n1 ≝ (n2-d) #IHd
+@IH -IH [| // ] #m #Hn
+/4 width=3 by lt_to_le, lt_to_le_to_lt/
+qed-.
+
(* Iterators ****************************************************************)
(* Note: see also: lib/arithemetics/bigops.ma *)
lemma iter_S: ∀B:Type[0]. ∀f:B→B.∀b,l. f^(S l) b = f (f^l b).
// qed.
-lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
-#B #f #b #l >commutative_plus //
-qed.
-
lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
#B #f #b #l elim l -l normalize //
qed.
| #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/
]
qed.
+
+(* Decidability of predicates ***********************************************)
+
+lemma dec_lt (R:predicate nat):
+ (∀n. Decidable … (R n)) →
+ ∀n. Decidable … (∃∃m. m < n & R m).
+#R #HR #n elim n -n [| #n * ]
+[ @or_intror * /2 width=2 by lt_zero_false/
+| * /4 width=3 by lt_S, or_introl, ex2_intro/
+| #H0 elim (HR n) -HR
+ [ /3 width=3 by or_introl, ex2_intro/
+ | #Hn @or_intror * #m #Hmn #Hm
+ elim (le_to_or_lt_eq … Hmn) -Hmn #H destruct [ -Hn | -H0 ]
+ /4 width=3 by lt_S_S_to_lt, ex2_intro/
+ ]
+]
+qed-.
+
+lemma dec_min (R:predicate nat):
+ (∀n. Decidable … (R n)) → ∀n. R n →
+ ∃∃m. m ≤ n & R m & (∀p. p < m → R p → ⊥).
+#R #HR #n
+@(nat_elim1 n) -n #n #IH #Hn
+elim (dec_lt … HR n) -HR [ -Hn | -IH ]
+[ * #p #Hpn #Hp
+ elim (IH … Hpn Hp) -IH -Hp #m #Hmp #Hm #HNm
+ @(ex3_intro … Hm HNm) -HNm
+ /3 width=3 by lt_to_le, le_to_lt_to_lt/
+| /4 width=4 by ex3_intro, ex2_intro/
+]
+qed-.
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