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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nneom 47401 | A positive integer is even or odd. (Contributed by AV, 30-May-2020.) |
β’ (π β β β ((π / 2) β β β¨ ((π β 1) / 2) β β0)) | ||
Theorem | nn0eo 47402 | A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.) |
β’ (π β β0 β ((π / 2) β β0 β¨ ((π + 1) / 2) β β0)) | ||
Theorem | nnpw2even 47403 | 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) |
β’ (π β β β ((2βπ) / 2) β β) | ||
Theorem | zefldiv2 47404 | The floor of an even integer divided by 2 is equal to the integer divided by 2. (Contributed by AV, 7-Jun-2020.) |
β’ ((π β β€ β§ (π / 2) β β€) β (ββ(π / 2)) = (π / 2)) | ||
Theorem | zofldiv2 47405 | The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) |
β’ ((π β β€ β§ ((π + 1) / 2) β β€) β (ββ(π / 2)) = ((π β 1) / 2)) | ||
Theorem | nn0ofldiv2 47406 | The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) (Proof shortened by AV, 7-Jun-2020.) |
β’ ((π β β0 β§ ((π + 1) / 2) β β0) β (ββ(π / 2)) = ((π β 1) / 2)) | ||
Theorem | flnn0div2ge 47407 | The floor of a positive integer divided by 2 is greater than or equal to the integer decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) |
β’ (π β β0 β ((π β 1) / 2) β€ (ββ(π / 2))) | ||
Theorem | flnn0ohalf 47408 | The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012.) |
β’ ((π β β0 β§ ((π + 1) / 2) β β0) β (ββ(π / 2)) = (ββ((π β 1) / 2))) | ||
Theorem | logcxp0 47409 | Logarithm of a complex power. Generalization of logcxp 26519. (Contributed by AV, 22-May-2020.) |
β’ ((π΄ β (β β {0}) β§ π΅ β β β§ (π΅ Β· (logβπ΄)) β ran log) β (logβ(π΄βππ΅)) = (π΅ Β· (logβπ΄))) | ||
Theorem | regt1loggt0 47410 | The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020.) |
β’ (π΅ β (1(,)+β) β 0 < (logβπ΅)) | ||
Syntax | cfdiv 47411 | Extend class notation with the division operator of two functions. |
class /f | ||
Definition | df-fdiv 47412* | Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.) |
β’ /f = (π β V, π β V β¦ ((π βf / π) βΎ (π supp 0))) | ||
Theorem | fdivval 47413 | The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.) |
β’ ((πΉ β π β§ πΊ β π) β (πΉ /f πΊ) = ((πΉ βf / πΊ) βΎ (πΊ supp 0))) | ||
Theorem | fdivmpt 47414* | The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.) |
β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ) = (π₯ β (πΊ supp 0) β¦ ((πΉβπ₯) / (πΊβπ₯)))) | ||
Theorem | fdivmptf 47415 | The quotient of two functions into the complex numbers is a function into the complex numbers. (Contributed by AV, 16-May-2020.) |
β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ):(πΊ supp 0)βΆβ) | ||
Theorem | refdivmptf 47416 | The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.) |
β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ):(πΊ supp 0)βΆβ) | ||
Theorem | fdivpm 47417 | The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.) |
β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ) β (β βpm π΄)) | ||
Theorem | refdivpm 47418 | The quotient of two functions into the real numbers is a partial function. (Contributed by AV, 16-May-2020.) |
β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ) β (β βpm π΄)) | ||
Theorem | fdivmptfv 47419 | The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.) |
β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β ((πΉ /f πΊ)βπ) = ((πΉβπ) / (πΊβπ))) | ||
Theorem | refdivmptfv 47420 | The function value of a quotient of two functions into the real numbers. (Contributed by AV, 19-May-2020.) |
β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β ((πΉ /f πΊ)βπ) = ((πΉβπ) / (πΊβπ))) | ||
Syntax | cbigo 47421 | Extend class notation with the class of the "big-O" function. |
class Ξ | ||
Definition | df-bigo 47422* | Define the function "big-O", mapping a real function g to the set of real functions "of order g(x)". Definition in section 1.1 of [AhoHopUll] p. 2. This is a generalization of "big-O of one", see df-o1 15431 and df-lo1 15432. As explained in the comment of df-o1 , any big-O can be represented in terms of π(1) and division, see elbigolo1 47431. (Contributed by AV, 15-May-2020.) |
β’ Ξ = (π β (β βpm β) β¦ {π β (β βpm β) β£ βπ₯ β β βπ β β βπ¦ β (dom π β© (π₯[,)+β))(πβπ¦) β€ (π Β· (πβπ¦))}) | ||
Theorem | bigoval 47423* | Set of functions of order G(x). (Contributed by AV, 15-May-2020.) |
β’ (πΊ β (β βpm β) β (ΞβπΊ) = {π β (β βpm β) β£ βπ₯ β β βπ β β βπ¦ β (dom π β© (π₯[,)+β))(πβπ¦) β€ (π Β· (πΊβπ¦))}) | ||
Theorem | elbigofrcl 47424 | Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.) |
β’ (πΉ β (ΞβπΊ) β πΊ β (β βpm β)) | ||
Theorem | elbigo 47425* | Properties of a function of order G(x). (Contributed by AV, 16-May-2020.) |
β’ (πΉ β (ΞβπΊ) β (πΉ β (β βpm β) β§ πΊ β (β βpm β) β§ βπ₯ β β βπ β β βπ¦ β (dom πΉ β© (π₯[,)+β))(πΉβπ¦) β€ (π Β· (πΊβπ¦)))) | ||
Theorem | elbigo2 47426* | Properties of a function of order G(x) under certain assumptions. (Contributed by AV, 17-May-2020.) |
β’ (((πΊ:π΄βΆβ β§ π΄ β β) β§ (πΉ:π΅βΆβ β§ π΅ β π΄)) β (πΉ β (ΞβπΊ) β βπ₯ β β βπ β β βπ¦ β π΅ (π₯ β€ π¦ β (πΉβπ¦) β€ (π Β· (πΊβπ¦))))) | ||
Theorem | elbigo2r 47427* | Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.) |
β’ (((πΊ:π΄βΆβ β§ π΄ β β) β§ (πΉ:π΅βΆβ β§ π΅ β π΄) β§ (πΆ β β β§ π β β β§ βπ₯ β π΅ (πΆ β€ π₯ β (πΉβπ₯) β€ (π Β· (πΊβπ₯))))) β πΉ β (ΞβπΊ)) | ||
Theorem | elbigof 47428 | A function of order G(x) is a function. (Contributed by AV, 18-May-2020.) |
β’ (πΉ β (ΞβπΊ) β πΉ:dom πΉβΆβ) | ||
Theorem | elbigodm 47429 | The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.) |
β’ (πΉ β (ΞβπΊ) β dom πΉ β β) | ||
Theorem | elbigoimp 47430* | The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.) |
β’ ((πΉ β (ΞβπΊ) β§ πΉ:π΄βΆβ β§ π΄ β dom πΊ) β βπ₯ β β βπ β β βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ¦) β€ (π Β· (πΊβπ¦)))) | ||
Theorem | elbigolo1 47431 | A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.) |
β’ ((π΄ β β β§ πΊ:π΄βΆβ+ β§ πΉ:π΄βΆβ+) β (πΉ β (ΞβπΊ) β (πΉ /f πΊ) β β€π(1))) | ||
Theorem | rege1logbrege0 47432 | The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.) |
β’ ((π΅ β (1(,)+β) β§ π β (1[,)+β)) β 0 β€ (π΅ logb π)) | ||
Theorem | rege1logbzge0 47433 | The general logarithm, with an integer base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β (1[,)+β)) β 0 β€ (π΅ logb π)) | ||
Theorem | fllogbd 47434 | A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.) |
β’ (π β π΅ β (β€β₯β2)) & β’ (π β π β β+) & β’ πΈ = (ββ(π΅ logb π)) β β’ (π β ((π΅βπΈ) β€ π β§ π < (π΅β(πΈ + 1)))) | ||
Theorem | relogbmulbexp 47435 | The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.) |
β’ ((π΅ β (β+ β {1}) β§ (π΄ β β+ β§ πΆ β β)) β (π΅ logb (π΄ Β· (π΅βππΆ))) = ((π΅ logb π΄) + πΆ)) | ||
Theorem | relogbdivb 47436 | The logarithm of the quotient of a positive real number and the base is the logarithm of the number minus 1. (Contributed by AV, 29-May-2020.) |
β’ ((π΅ β (β+ β {1}) β§ π΄ β β+) β (π΅ logb (π΄ / π΅)) = ((π΅ logb π΄) β 1)) | ||
Theorem | logbge0b 47437 | The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β β+) β (0 β€ (π΅ logb π) β 1 β€ π)) | ||
Theorem | logblt1b 47438 | The logarithm of a number is less than 1 iff the number is less than the base of the logarithm. (Contributed by AV, 30-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β β+) β ((π΅ logb π) < 1 β π < π΅)) | ||
If the binary logarithm is used more often, a separate symbol/definition could be provided for it, e.g., log2 = (π₯ β (β β {0}) β¦ (2 logb π)). Then we can write "( log2 ` x )" (analogous to (logπ₯) for the natural logarithm) instead of (2 logb π₯). | ||
Theorem | fldivexpfllog2 47439 | The floor of a positive real number divided by 2 to the power of the floor of the logarithm to base 2 of the number is 1. (Contributed by AV, 26-May-2020.) |
β’ (π β β+ β (ββ(π / (2β(ββ(2 logb π))))) = 1) | ||
Theorem | nnlog2ge0lt1 47440 | A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.) |
β’ (π β β β (π = 1 β (0 β€ (2 logb π) β§ (2 logb π) < 1))) | ||
Theorem | logbpw2m1 47441 | The floor of the binary logarithm of 2 to the power of a positive integer minus 1 is equal to the integer minus 1. (Contributed by AV, 31-May-2020.) |
β’ (πΌ β β β (ββ(2 logb ((2βπΌ) β 1))) = (πΌ β 1)) | ||
Theorem | fllog2 47442 | The floor of the binary logarithm of 2 to the power of an element of a half-open integer interval bounded by powers of 2 is equal to the integer. (Contributed by AV, 31-May-2020.) |
β’ ((πΌ β β0 β§ π β ((2βπΌ)..^(2β(πΌ + 1)))) β (ββ(2 logb π)) = πΌ) | ||
Syntax | cblen 47443 | Extend class notation with the class of the binary length function. |
class #b | ||
Definition | df-blen 47444 | Define the binary length of an integer. Definition in section 1.3 of [AhoHopUll] p. 12. Although not restricted to integers, this definition is only meaningful for π β β€ or even for π β β. (Contributed by AV, 16-May-2020.) |
β’ #b = (π β V β¦ if(π = 0, 1, ((ββ(2 logb (absβπ))) + 1))) | ||
Theorem | blenval 47445 | The binary length of an integer. (Contributed by AV, 20-May-2020.) |
β’ (π β π β (#bβπ) = if(π = 0, 1, ((ββ(2 logb (absβπ))) + 1))) | ||
Theorem | blen0 47446 | The binary length of 0. (Contributed by AV, 20-May-2020.) |
β’ (#bβ0) = 1 | ||
Theorem | blenn0 47447 | The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.) |
β’ ((π β π β§ π β 0) β (#bβπ) = ((ββ(2 logb (absβπ))) + 1)) | ||
Theorem | blenre 47448 | The binary length of a positive real number. (Contributed by AV, 20-May-2020.) |
β’ (π β β+ β (#bβπ) = ((ββ(2 logb π)) + 1)) | ||
Theorem | blennn 47449 | The binary length of a positive integer. (Contributed by AV, 21-May-2020.) |
β’ (π β β β (#bβπ) = ((ββ(2 logb π)) + 1)) | ||
Theorem | blennnelnn 47450 | The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.) |
β’ (π β β β (#bβπ) β β) | ||
Theorem | blennn0elnn 47451 | The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.) |
β’ (π β β0 β (#bβπ) β β) | ||
Theorem | blenpw2 47452 | The binary length of a power of 2 is the exponent plus 1. (Contributed by AV, 30-May-2020.) |
β’ (πΌ β β0 β (#bβ(2βπΌ)) = (πΌ + 1)) | ||
Theorem | blenpw2m1 47453 | The binary length of a power of 2 minus 1 is the exponent. (Contributed by AV, 31-May-2020.) |
β’ (πΌ β β β (#bβ((2βπΌ) β 1)) = πΌ) | ||
Theorem | nnpw2blen 47454 | A positive integer is between 2 to the power of its binary length minus 1 and 2 to the power of its binary length. (Contributed by AV, 31-May-2020.) |
β’ (π β β β ((2β((#bβπ) β 1)) β€ π β§ π < (2β(#bβπ)))) | ||
Theorem | nnpw2blenfzo 47455 | A positive integer is between 2 to the power of the binary length of the integer minus 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.) |
β’ (π β β β π β ((2β((#bβπ) β 1))..^(2β(#bβπ)))) | ||
Theorem | nnpw2blenfzo2 47456 | A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.) |
β’ (π β β β (π = (2β((#bβπ) β 1)) β¨ π β (((2β((#bβπ) β 1)) + 1)..^(2β(#bβπ))))) | ||
Theorem | nnpw2pmod 47457 | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
β’ (π β β β π = ((2β((#bβπ) β 1)) + (π mod (2β((#bβπ) β 1))))) | ||
Theorem | blen1 47458 | The binary length of 1. (Contributed by AV, 21-May-2020.) |
β’ (#bβ1) = 1 | ||
Theorem | blen2 47459 | The binary length of 2. (Contributed by AV, 21-May-2020.) |
β’ (#bβ2) = 2 | ||
Theorem | nnpw2p 47460* | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
β’ (π β β β βπ β β0 βπ β (0..^(2βπ))π = ((2βπ) + π)) | ||
Theorem | nnpw2pb 47461* | A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
β’ (π β β β βπ β β0 βπ β (0..^(2βπ))π = ((2βπ) + π)) | ||
Theorem | blen1b 47462 | The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.) |
β’ (π β β0 β ((#bβπ) = 1 β (π = 0 β¨ π = 1))) | ||
Theorem | blennnt2 47463 | The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.) |
β’ (π β β β (#bβ(2 Β· π)) = ((#bβπ) + 1)) | ||
Theorem | nnolog2flm1 47464 | The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.) |
β’ ((π β (β€β₯β2) β§ ((π + 1) / 2) β β) β (ββ(2 logb π)) = (ββ(2 logb (π β 1)))) | ||
Theorem | blennn0em1 47465 | The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.) |
β’ ((π β β β§ (π / 2) β β0) β (#bβ(π / 2)) = ((#bβπ) β 1)) | ||
Theorem | blennngt2o2 47466 | The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.) |
β’ ((π β (β€β₯β2) β§ ((π + 1) / 2) β β0) β (#bβπ) = ((#bβ((π β 1) / 2)) + 1)) | ||
Theorem | blengt1fldiv2p1 47467 | The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.) |
β’ (π β (β€β₯β2) β (#bβπ) = ((#bβ(ββ(π / 2))) + 1)) | ||
Theorem | blennn0e2 47468 | The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.) |
β’ ((π β β β§ (π / 2) β β0) β (#bβπ) = ((#bβ(π / 2)) + 1)) | ||
Generalization of df-bits 16360. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 47488: if πΎ and π are nonnegative integers, then ((πΎ(digitβ2)π) = 1 β πΎ β (bitsβπ)). | ||
Syntax | cdig 47469 | Extend class notation with the class of the digit extraction operation. |
class digit | ||
Definition | df-dig 47470* | Definition of an operation to obtain the π th digit of a nonnegative real number π in the positional system with base π. π = β 1 corresponds to the first digit of the fractional part (for π = 10 the first digit after the decimal point), π = 0 corresponds to the last digit of the integer part (for π = 10 the first digit before the decimal point). See also digit1 14197. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.) |
β’ digit = (π β β β¦ (π β β€, π β (0[,)+β) β¦ ((ββ((πβ-π) Β· π)) mod π))) | ||
Theorem | digfval 47471* | Operation to obtain the π th digit of a nonnegative real number π in the positional system with base π΅. (Contributed by AV, 23-May-2020.) |
β’ (π΅ β β β (digitβπ΅) = (π β β€, π β (0[,)+β) β¦ ((ββ((π΅β-π) Β· π)) mod π΅))) | ||
Theorem | digval 47472 | The πΎ th digit of a nonnegative real number π in the positional system with base π΅. (Contributed by AV, 23-May-2020.) |
β’ ((π΅ β β β§ πΎ β β€ β§ π β (0[,)+β)) β (πΎ(digitβπ΅)π ) = ((ββ((π΅β-πΎ) Β· π )) mod π΅)) | ||
Theorem | digvalnn0 47473 | The πΎ th digit of a nonnegative real number π in the positional system with base π΅ is a nonnegative integer. (Contributed by AV, 28-May-2020.) |
β’ ((π΅ β β β§ πΎ β β€ β§ π β (0[,)+β)) β (πΎ(digitβπ΅)π ) β β0) | ||
Theorem | nn0digval 47474 | The πΎ th digit of a nonnegative real number π in the positional system with base π΅. (Contributed by AV, 23-May-2020.) |
β’ ((π΅ β β β§ πΎ β β0 β§ π β (0[,)+β)) β (πΎ(digitβπ΅)π ) = ((ββ(π / (π΅βπΎ))) mod π΅)) | ||
Theorem | dignn0fr 47475 | The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.) |
β’ ((π΅ β β β§ πΎ β (β€ β β0) β§ π β β0) β (πΎ(digitβπ΅)π) = 0) | ||
Theorem | dignn0ldlem 47476 | Lemma for dignnld 47477. (Contributed by AV, 25-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β β β§ πΎ β (β€β₯β((ββ(π΅ logb π)) + 1))) β π < (π΅βπΎ)) | ||
Theorem | dignnld 47477 | The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β β β§ πΎ β (β€β₯β((ββ(π΅ logb π)) + 1))) β (πΎ(digitβπ΅)π) = 0) | ||
Theorem | dig2nn0ld 47478 | The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.) |
β’ ((π β β β§ πΎ β (β€β₯β(#bβπ))) β (πΎ(digitβ2)π) = 0) | ||
Theorem | dig2nn1st 47479 | The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.) |
β’ (π β β β (((#bβπ) β 1)(digitβ2)π) = 1) | ||
Theorem | dig0 47480 | All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
β’ ((π΅ β β β§ πΎ β β€) β (πΎ(digitβπ΅)0) = 0) | ||
Theorem | digexp 47481 | The πΎ th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ πΎ β β0 β§ π β β0) β (πΎ(digitβπ΅)(π΅βπ)) = if(πΎ = π, 1, 0)) | ||
Theorem | dig1 47482 | All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ πΎ β β€) β (πΎ(digitβπ΅)1) = if(πΎ = 0, 1, 0)) | ||
Theorem | 0dig1 47483 | The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.) |
β’ (π΅ β (β€β₯β2) β (0(digitβπ΅)1) = 1) | ||
Theorem | 0dig2pr01 47484 | The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.) |
β’ (π β {0, 1} β (0(digitβ2)π) = π) | ||
Theorem | dig2nn0 47485 | A digit of a nonnegative integer π in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.) |
β’ ((π β β0 β§ πΎ β β€) β (πΎ(digitβ2)π) β {0, 1}) | ||
Theorem | 0dig2nn0e 47486 | The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.) |
β’ ((π β β0 β§ (π / 2) β β0) β (0(digitβ2)π) = 0) | ||
Theorem | 0dig2nn0o 47487 | The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.) |
β’ ((π β β0 β§ ((π + 1) / 2) β β0) β (0(digitβ2)π) = 1) | ||
Theorem | dig2bits 47488 | The πΎ th digit of a nonnegative integer π in a binary system is its πΎ th bit. (Contributed by AV, 24-May-2020.) |
β’ ((π β β0 β§ πΎ β β0) β ((πΎ(digitβ2)π) = 1 β πΎ β (bitsβπ))) | ||
Theorem | dignn0flhalflem1 47489 | Lemma 1 for dignn0flhalf 47492. (Contributed by AV, 7-Jun-2012.) |
β’ ((π΄ β β€ β§ ((π΄ β 1) / 2) β β β§ π β β) β (ββ((π΄ / (2βπ)) β 1)) < (ββ((π΄ β 1) / (2βπ)))) | ||
Theorem | dignn0flhalflem2 47490 | Lemma 2 for dignn0flhalf 47492. (Contributed by AV, 7-Jun-2012.) |
β’ ((π΄ β β€ β§ ((π΄ β 1) / 2) β β β§ π β β0) β (ββ(π΄ / (2β(π + 1)))) = (ββ((ββ(π΄ / 2)) / (2βπ)))) | ||
Theorem | dignn0ehalf 47491 | The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.) |
β’ (((π΄ / 2) β β0 β§ π΄ β β0 β§ πΌ β β0) β ((πΌ + 1)(digitβ2)π΄) = (πΌ(digitβ2)(π΄ / 2))) | ||
Theorem | dignn0flhalf 47492 | The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.) |
β’ ((π΄ β (β€β₯β2) β§ πΌ β β0) β ((πΌ + 1)(digitβ2)π΄) = (πΌ(digitβ2)(ββ(π΄ / 2)))) | ||
Theorem | nn0sumshdiglemA 47493* | Lemma for nn0sumshdig 47497 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.) |
β’ (((π β β β§ (π / 2) β β) β§ π¦ β β) β (βπ₯ β β0 ((#bβπ₯) = π¦ β π₯ = Ξ£π β (0..^π¦)((π(digitβ2)π₯) Β· (2βπ))) β ((#bβπ) = (π¦ + 1) β π = Ξ£π β (0..^(π¦ + 1))((π(digitβ2)π) Β· (2βπ))))) | ||
Theorem | nn0sumshdiglemB 47494* | Lemma for nn0sumshdig 47497 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.) |
β’ (((π β β β§ ((π β 1) / 2) β β0) β§ π¦ β β) β (βπ₯ β β0 ((#bβπ₯) = π¦ β π₯ = Ξ£π β (0..^π¦)((π(digitβ2)π₯) Β· (2βπ))) β ((#bβπ) = (π¦ + 1) β π = Ξ£π β (0..^(π¦ + 1))((π(digitβ2)π) Β· (2βπ))))) | ||
Theorem | nn0sumshdiglem1 47495* | Lemma 1 for nn0sumshdig 47497 (induction step). (Contributed by AV, 7-Jun-2020.) |
β’ (π¦ β β β (βπ β β0 ((#bβπ) = π¦ β π = Ξ£π β (0..^π¦)((π(digitβ2)π) Β· (2βπ))) β βπ β β0 ((#bβπ) = (π¦ + 1) β π = Ξ£π β (0..^(π¦ + 1))((π(digitβ2)π) Β· (2βπ))))) | ||
Theorem | nn0sumshdiglem2 47496* | Lemma 2 for nn0sumshdig 47497. (Contributed by AV, 7-Jun-2020.) |
β’ (πΏ β β β βπ β β0 ((#bβπ) = πΏ β π = Ξ£π β (0..^πΏ)((π(digitβ2)π) Β· (2βπ)))) | ||
Theorem | nn0sumshdig 47497* | A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.) |
β’ (π΄ β β0 β π΄ = Ξ£π β (0..^(#bβπ΄))((π(digitβ2)π΄) Β· (2βπ))) | ||
Theorem | nn0mulfsum 47498* | Trivial algorithm to calculate the product of two nonnegative integers π and π by adding π to itself π times. (Contributed by AV, 17-May-2020.) |
β’ ((π΄ β β0 β§ π΅ β β0) β (π΄ Β· π΅) = Ξ£π β (1...π΄)π΅) | ||
Theorem | nn0mullong 47499* | Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers π and π by multiplying the multiplicand π by each digit of the multiplier π and then add up all the properly shifted results. Here, the binary representation of the multiplier π is used, i.e., the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 16431. (Contributed by AV, 7-Jun-2020.) |
β’ ((π΄ β β0 β§ π΅ β β0) β (π΄ Β· π΅) = Ξ£π β (0..^(#bβπ΄))(((π(digitβ2)π΄) Β· (2βπ)) Β· π΅)) | ||
According to Wikipedia ("Arity", https://en.wikipedia.org/wiki/Arity, 19-May-2024): "In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation." N-ary functions are often also called multivariate functions, without indicating the actual number of argumens. See also Wikipedia ("Multivariate functions", 19-May-2024, https://en.wikipedia.org/wiki/Function_(mathematics)#Multivariate_functions ): "A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. ... Formally, a function of n variables is a function whose domain is a set of n-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. Commonly, an n-tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , ... , n ). When using functional notation, one usually omits the parentheses surrounding tuples, writing f ( x1 , ... , xn ) instead of f ( ( x1 , ... , xn ) ). Given n sets X1 , ... , Xn , the set of all n-tuples ( x1 , ... , xn ) such that x1 is element of X1 , ... , xn is element of Xn is called the Cartesian product of X1 , ... , Xn , and denoted X1 X ... X Xn . Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain: π:πβΆπ where where the domain π has the form π β ((...((πβ1) Γ (πβ2)) Γ ...) Γ (πβπ))." In the following, n-ary functions are defined as mappings (see df-map 8818) from a finite sequence of arguments, which themselves are defined as mappings from the half-open range of nonnegative integers to the domain of each argument. Furthermore, the definition is restricted to endofunctions, meaning that the domain(s) of the argument(s) is identical with its codomain. This means that the domains of all arguments are identical (in contrast to the definition in Wikipedia, see above: here, we have X1 = X2 = ... = Xn = X). For small n, n-ary functions correspond to "usual" functions with a different number of arguments: - n = 0 (nullary functions): These correspond actually to constants, see 0aryfvalelfv 47509 and mapsn 8878: (π βm {β }) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 47519 and efmndbas 18786: (π βm π) - n = 2 (binary functions): These correspond to usual operations on two elements of the same set, also called "binary operation" (according to Wikipedia ("Binary operation", 19-May-2024, https://en.wikipedia.org/wiki/Binary_operation 18786): "In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is a binary operation whose two domains and the codomain are the same set." Sometimes also called "closed internal binary operation"), see 2aryenef 47530 and compare with df-clintop 47063: (π βm (π Γ π)). Instead of using indexed arguments (represented by a mapping as described above), elements of Cartesian exponentiations (πββπ) (see df-finxp 36755) could have been used to represent multiple arguments. However, this concept is not fully developed yet (it is within a mathbox), and it is currently based on ordinal numbers, e.g., (πββ2o), instead of integers, e.g., (πββ2), which is not very practical. The definition df-ixp of infinite Cartesian product could also have been used to represent multiple arguments, but this would have been more cumbersome without any additional advantage. naryfvalixp 47503 shows that both definitions are equivalent. | ||
Syntax | cnaryf 47500 | Extend the definition of a class to include the n-ary functions. |
class -aryF |
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