exp and factorial

[helm.git] / matita / matita / lib / arithmetics / exp.ma

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+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic        
+    ||A||  Library of Mathematics, developed at the Computer Science     
+    ||T||  Department of the University of Bologna, Italy.                     
+    ||I||                                                                 
+    ||T||  
+    ||A||  This file is distributed under the terms of the 
+    \   /  GNU General Public License Version 2        
+     \ /      
+      V_______________________________________________________________ *)
+
+include "arithmetics/div_and_mod.ma".
+
+let rec exp n m on m ≝ 
+ match m with 
+ [ O ⇒ 1
+ | S p ⇒ (exp n p) * n].
+
+interpretation "natural exponent" 'exp a b = (exp a b).
+
+theorem exp_plus_times : ∀n,p,q:nat. 
+  n^(p + q) = n^p * n^q.
+#n #p #q elim p normalize //
+qed.
+
+theorem exp_n_O : ∀n:nat. 1 = n^O. 
+//
+qed.
+
+theorem exp_n_1 : ∀n:nat. n = n^1. 
+#n normalize //
+qed.
+
+theorem exp_1_n : ∀n:nat. 1 = 1^n.
+#n (elim n) normalize //
+qed.
+
+theorem exp_2: ∀n. n^2 = n*n. 
+#n normalize //
+qed.
+
+theorem exp_exp_times : ∀n,p,q:nat. 
+  (n^p)^q = n^(p * q).
+#n #p #q (elim q) normalize 
+(* [applyS exp_n_O funziona ma non lo trova *)
+// <times_n_O // 
+qed.
+
+theorem lt_O_exp: ∀n,m:nat. O < n → O < n^m. 
+#n #m (elim m) normalize // #a #Hind #posn 
+@(le_times 1 ? 1) /2/
+qed.
+
+theorem lt_m_exp_nm: ∀n,m:nat. 1 < n → m < n^m.
+#n #m #lt1n (elim m) normalize // 
+#n #Hind @(transitive_le ? ((S n)*2)) // @le_times //
+qed.
+
+theorem exp_to_eq_O: ∀n,m:nat. 1 < n → 
+  n^m = 1 → m = O.
+#n #m #ltin #eq1 @le_to_le_to_eq //
+@le_S_S_to_le <eq1 @lt_m_exp_nm //
+qed.
+
+theorem injective_exp_r: ∀b:nat. 1 < b → 
+  injective nat nat (λi:nat. b^i).
+#b #lt1b @nat_elim2 normalize 
+  [#n #H @sym_eq @(exp_to_eq_O b n lt1b) //
+  |#n #H @False_ind @(absurd (1 < 1) ? (not_le_Sn_n 1))
+   <H in ⊢ (??%) @(lt_to_le_to_lt ? (1*b)) //
+   @le_times // @lt_O_exp /2/
+  |#n #m #Hind #H @eq_f @Hind @(injective_times_l … H) /2/
+  ]
+qed.
+
+theorem le_exp: ∀n,m,p:nat. O < p →
+  n ≤m → p^n ≤ p^m.
+@nat_elim2 
+  [#n #m #ltm #len @lt_O_exp //
+  |#n #m #_ #len @False_ind /2/
+  |#n #m #Hind #p #posp #lenm normalize @le_times //
+   @Hind /2/
+  ]
+qed.
+
+theorem le_exp1: ∀n,m,a:nat. O < a →
+  n ≤m → n^a ≤ m^a.
+#n #m #a #posa #lenm (elim posa) //
+#a #posa #Hind @le_times //
+qed.
+
+theorem lt_exp: ∀n,m,p:nat. 1 < p → 
+  n < m → p^n < p^m.
+#n #m #p #lt1p #ltnm 
+cut (p \sup n ≤ p \sup m) [@le_exp /2/] #H 
+cases(le_to_or_lt_eq … H) // #eqexp
+@False_ind @(absurd (n=m)) /2/
+qed.
+
+theorem lt_exp1: ∀n,m,p:nat. 0 < p → 
+  n < m → n^p < m^p.
+#n #m #p #posp #ltnm (elim posp) //
+#p #posp #Hind @lt_times //
+qed.
+  
+theorem le_exp_to_le: 
+∀b,n,m. 1 < b → b^n ≤ b^m → n ≤ m.
+#b #n #m #lt1b #leexp cases(decidable_le n m) //
+#notle @False_ind @(absurd … leexp) @lt_to_not_le
+@lt_exp /2/
+qed.
+
+theorem le_exp_to_le1 : ∀n,m,p.O < p → 
+  n^p ≤ m^p → n ≤ m.
+#n #m #p #posp #leexp @not_lt_to_le 
+@(not_to_not … (lt_exp1 ??? posp)) @le_to_not_lt // 
+qed.
+     
+theorem lt_exp_to_lt: 
+∀a,n,m. 0 < a → a^n < a^m → n < m.
+#a #n #m #lt1a #ltexp cases(decidable_le (S n) m) //
+#H @False_ind @(absurd … ltexp) @le_to_not_lt 
+@le_exp // @not_lt_to_le @H
+qed.
+
+theorem lt_exp_to_lt1: 
+∀a,n,m. O < a → n^a < m^a → n < m.
+#a #n #m #posa #ltexp cases(decidable_le (S n) m) //
+#H @False_ind @(absurd … ltexp) @le_to_not_lt 
+@le_exp1 // @not_lt_to_le @H 
+qed.
+     
+theorem times_exp: ∀n,m,p. 
+  n^p * m^p = (n*m)^p.
+#n #m #p (elim p) // #p #Hind normalize //
+qed.
+
+  
+   
+   
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