Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

### The abstract of the Mizar article:

### Natural Numbers

**by****Robert Milewski**- Received February 23, 1998
- MML identifier: NAT_2

- [ Mizar article, MML identifier index ]

environ vocabulary FINSEQ_1, FUNCT_1, INT_1, ARYTM_1, ARYTM_3, NAT_1, POWER, MATRIX_2, FINSEQ_4, ORDINAL2, REALSET1, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XCMPLX_0, XREAL_0, REALSET1, INT_1, NAT_1, POWER, ABIAN, SERIES_1, FUNCT_1, FINSEQ_1, FINSEQ_4, BINARITH, ORDINAL2; constructors ABIAN, SERIES_1, BINARITH, FINSEQ_4, INT_1, REALSET1, MEMBERED; clusters XREAL_0, RELSET_1, ABIAN, BINARITH, INT_1, REALSET1, MEMBERED, NAT_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Preliminaries scheme NonUniqPiFinRecExD{D() -> non empty set, A() -> Element of D(), N() -> Nat, P[set,set,set]}: ex p be FinSequence of D() st len p = N() & (p/.1 = A() or N() = 0) & for n be Nat st 1 <= n & n < N() holds P[n,p/.n,p/.(n+1)] provided for n be Nat st 1 <= n & n < N() holds for x be Element of D() ex y be Element of D() st P[n,x,y]; theorem :: NAT_2:1 for x be real number holds x < [\ x /] + 1; theorem :: NAT_2:2 for x,y be real number st x >= 0 & y > 0 holds x / ( [\ x / y /] + 1 ) < y; begin :: Division and Rest of Division canceled; theorem :: NAT_2:4 for n be natural number holds 0 div n = 0; theorem :: NAT_2:5 for n be non empty Nat holds n div n = 1; theorem :: NAT_2:6 for n be Nat holds n div 1 = n; theorem :: NAT_2:7 for i,j,k,l be natural number st i <= j & k <= j holds i = j -' k + l implies k = j -' i + l; theorem :: NAT_2:8 for i,n be natural number holds i in Seg n implies n -' i + 1 in Seg n; theorem :: NAT_2:9 for i,j be natural number st j < i holds i -' (j + 1) + 1 = i -' j; theorem :: NAT_2:10 for i,j be natural number st i >= j holds j -' i = 0; theorem :: NAT_2:11 for i,j be non empty natural number holds i -' j < i; theorem :: NAT_2:12 for n,k be natural number st k <= n holds 2 to_power n = (2 to_power k) * (2 to_power (n-'k)); theorem :: NAT_2:13 for n,k be Nat st k <= n holds 2 to_power k divides 2 to_power n; theorem :: NAT_2:14 for n,k be natural number st k > 0 & n div k = 0 holds n < k; reserve n, k, i for natural number; theorem :: NAT_2:15 k > 0 & k <= n implies n div k >= 1; theorem :: NAT_2:16 k <> 0 implies (n+k) div k = (n div k) + 1; theorem :: NAT_2:17 k divides n & 1 <= n & 1 <= i & i <= k implies (n -' i) div k = (n div k) - 1; theorem :: NAT_2:18 for n,k be Nat st k <= n holds (2 to_power n) div (2 to_power k) = 2 to_power (n -' k); theorem :: NAT_2:19 for n be Nat st n > 0 holds 2 to_power n mod 2 = 0; theorem :: NAT_2:20 for n be Nat st n > 0 holds n mod 2 = 0 iff (n -' 1) mod 2 = 1; theorem :: NAT_2:21 for n be non empty natural number st n <> 1 holds n > 1; theorem :: NAT_2:22 for n, k be natural number st n <= k & k < n + n holds k div n = 1; theorem :: NAT_2:23 for n be Nat holds n is even iff n mod 2 = 0; theorem :: NAT_2:24 for n be Nat holds n is odd iff n mod 2 = 1; theorem :: NAT_2:25 for n,k,t be Nat st 1 <= t & k <= n & 2*t divides k holds n div t is even iff (n-'k) div t is even; theorem :: NAT_2:26 for n,m,k be Nat st n <= m holds n div k <= m div k; theorem :: NAT_2:27 for n,k be Nat st k <= 2 * n holds (k+1) div 2 <= n; theorem :: NAT_2:28 for n be even Nat holds n div 2 = (n + 1) div 2; theorem :: NAT_2:29 for n,k,i be Nat holds (n div k) div i = n div (k*i); definition let n be natural number; redefine attr n is trivial means :: NAT_2:def 1 n = 0 or n = 1; end; definition cluster non trivial Nat; cluster non trivial natural number; end; theorem :: NAT_2:30 for k be natural number holds k is non trivial iff k is non empty & k <> 1; theorem :: NAT_2:31 for k be non trivial natural number holds k >= 2; scheme Ind_from_2 { P[Nat] } : for k be non trivial Nat holds P[k] provided P[2] and for k be non trivial Nat st P[k] holds P[k + 1];

Back to top