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Type | Label | Description |
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Statement | ||
Theorem | dignn0flhalflem1 47601 | Lemma 1 for dignn0flhalf 47604. (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 47602 | Lemma 2 for dignn0flhalf 47604. (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 47603 | 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 47604 | 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 47605* | Lemma for nn0sumshdig 47609 (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 47606* | Lemma for nn0sumshdig 47609 (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 47607* | Lemma 1 for nn0sumshdig 47609 (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 47608* | Lemma 2 for nn0sumshdig 47609. (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 47609* | 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 47610* | 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 47611* | 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 16453. (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 8836) 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 47621 and mapsn 8896: (π βm {β }) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 47631 and efmndbas 18808: (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 18808): "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 47642 and compare with df-clintop 47175: (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 36786) 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 47615 shows that both definitions are equivalent. | ||
Syntax | cnaryf 47612 | Extend the definition of a class to include the n-ary functions. |
class -aryF | ||
Definition | df-naryf 47613* | Define the n-ary (endo)functions. (Contributed by AV, 11-May-2024.) (Revised by TA and SN, 7-Jun-2024.) |
β’ -aryF = (π β β0, π₯ β V β¦ (π₯ βm (π₯ βm (0..^π)))) | ||
Theorem | naryfval 47614 | The set of the n-ary (endo)functions on a class π. (Contributed by AV, 13-May-2024.) |
β’ πΌ = (0..^π) β β’ (π β β0 β (π-aryF π) = (π βm (π βm πΌ))) | ||
Theorem | naryfvalixp 47615* | The set of the n-ary (endo)functions on a class π expressed with the notation of infinite Cartesian products. (Contributed by AV, 19-May-2024.) |
β’ πΌ = (0..^π) β β’ (π β β0 β (π-aryF π) = (π βm Xπ₯ β πΌ π)) | ||
Theorem | naryfvalel 47616 | An n-ary (endo)function on a set π. (Contributed by AV, 14-May-2024.) |
β’ πΌ = (0..^π) β β’ ((π β β0 β§ π β π) β (πΉ β (π-aryF π) β πΉ:(π βm πΌ)βΆπ)) | ||
Theorem | naryrcl 47617 | Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.) |
β’ πΌ = (0..^π) β β’ (πΉ β (π-aryF π) β (π β β0 β§ π β V)) | ||
Theorem | naryfvalelfv 47618 | The value of an n-ary (endo)function on a set π is an element of π. (Contributed by AV, 14-May-2024.) |
β’ πΌ = (0..^π) β β’ ((πΉ β (π-aryF π) β§ π΄:πΌβΆπ) β (πΉβπ΄) β π) | ||
Theorem | naryfvalelwrdf 47619* | An n-ary (endo)function on a set π expressed as a function over the set of words on π of length π. (Contributed by AV, 4-Jun-2024.) |
β’ ((π β β0 β§ π β π) β (πΉ β (π-aryF π) β πΉ:{π€ β Word π β£ (β―βπ€) = π}βΆπ)) | ||
Theorem | 0aryfvalel 47620* | A nullary (endo)function on a set π is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: (πΉββ ) = πΆ with πΆ β π, see also 0aryfvalelfv 47621. Instead of (πΉββ ), nullary functions are usually written as πΉ() in literature. (Contributed by AV, 15-May-2024.) |
β’ (π β π β (πΉ β (0-aryF π) β βπ₯ β π πΉ = {β¨β , π₯β©})) | ||
Theorem | 0aryfvalelfv 47621* | The value of a nullary (endo)function on a set π. (Contributed by AV, 19-May-2024.) |
β’ (πΉ β (0-aryF π) β βπ₯ β π (πΉββ ) = π₯) | ||
Theorem | 1aryfvalel 47622 | A unary (endo)function on a set π. (Contributed by AV, 15-May-2024.) |
β’ (π β π β (πΉ β (1-aryF π) β πΉ:(π βm {0})βΆπ)) | ||
Theorem | fv1arycl 47623 | Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.) |
β’ ((πΊ β (1-aryF π) β§ π΄ β π) β (πΊβ{β¨0, π΄β©}) β π) | ||
Theorem | 1arympt1 47624* | A unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.) |
β’ πΉ = (π₯ β (π βm {0}) β¦ (π΄β(π₯β0))) β β’ ((π β π β§ π΄:πβΆπ) β πΉ β (1-aryF π)) | ||
Theorem | 1arympt1fv 47625* | The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.) |
β’ πΉ = (π₯ β (π βm {0}) β¦ (π΄β(π₯β0))) β β’ ((π β π β§ π΅ β π) β (πΉβ{β¨0, π΅β©}) = (π΄βπ΅)) | ||
Theorem | 1arymaptfv 47626* | The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) |
β’ π» = (β β (1-aryF π) β¦ (π₯ β π β¦ (ββ{β¨0, π₯β©}))) β β’ (πΉ β (1-aryF π) β (π»βπΉ) = (π₯ β π β¦ (πΉβ{β¨0, π₯β©}))) | ||
Theorem | 1arymaptf 47627* | The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024.) |
β’ π» = (β β (1-aryF π) β¦ (π₯ β π β¦ (ββ{β¨0, π₯β©}))) β β’ (π β π β π»:(1-aryF π)βΆ(π βm π)) | ||
Theorem | 1arymaptf1 47628* | The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.) |
β’ π» = (β β (1-aryF π) β¦ (π₯ β π β¦ (ββ{β¨0, π₯β©}))) β β’ (π β π β π»:(1-aryF π)β1-1β(π βm π)) | ||
Theorem | 1arymaptfo 47629* | The mapping of unary (endo)functions is a function onto the set of endofunctions. (Contributed by AV, 18-May-2024.) |
β’ π» = (β β (1-aryF π) β¦ (π₯ β π β¦ (ββ{β¨0, π₯β©}))) β β’ (π β π β π»:(1-aryF π)βontoβ(π βm π)) | ||
Theorem | 1arymaptf1o 47630* | The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions. (Contributed by AV, 19-May-2024.) |
β’ π» = (β β (1-aryF π) β¦ (π₯ β π β¦ (ββ{β¨0, π₯β©}))) β β’ (π β π β π»:(1-aryF π)β1-1-ontoβ(π βm π)) | ||
Theorem | 1aryenef 47631 | The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.) |
β’ (1-aryF π) β (π βm π) | ||
Theorem | 1aryenefmnd 47632 | The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.) |
β’ (1-aryF π) β (Baseβ(EndoFMndβπ)) | ||
Theorem | 2aryfvalel 47633 | A binary (endo)function on a set π. (Contributed by AV, 20-May-2024.) |
β’ (π β π β (πΉ β (2-aryF π) β πΉ:(π βm {0, 1})βΆπ)) | ||
Theorem | fv2arycl 47634 | Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.) |
β’ ((πΊ β (2-aryF π) β§ π΄ β π β§ π΅ β π) β (πΊβ{β¨0, π΄β©, β¨1, π΅β©}) β π) | ||
Theorem | 2arympt 47635* | A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.) |
β’ πΉ = (π₯ β (π βm {0, 1}) β¦ ((π₯β0)π(π₯β1))) β β’ ((π β π β§ π:(π Γ π)βΆπ) β πΉ β (2-aryF π)) | ||
Theorem | 2arymptfv 47636* | The value of a binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.) |
β’ πΉ = (π₯ β (π βm {0, 1}) β¦ ((π₯β0)π(π₯β1))) β β’ ((π β π β§ π΄ β π β§ π΅ β π) β (πΉβ{β¨0, π΄β©, β¨1, π΅β©}) = (π΄ππ΅)) | ||
Theorem | 2arymaptfv 47637* | The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.) |
β’ π» = (β β (2-aryF π) β¦ (π₯ β π, π¦ β π β¦ (ββ{β¨0, π₯β©, β¨1, π¦β©}))) β β’ (πΉ β (2-aryF π) β (π»βπΉ) = (π₯ β π, π¦ β π β¦ (πΉβ{β¨0, π₯β©, β¨1, π¦β©}))) | ||
Theorem | 2arymaptf 47638* | The mapping of binary (endo)functions is a function into the set of binary operations. (Contributed by AV, 21-May-2024.) |
β’ π» = (β β (2-aryF π) β¦ (π₯ β π, π¦ β π β¦ (ββ{β¨0, π₯β©, β¨1, π¦β©}))) β β’ (π β π β π»:(2-aryF π)βΆ(π βm (π Γ π))) | ||
Theorem | 2arymaptf1 47639* | The mapping of binary (endo)functions is a one-to-one function into the set of binary operations. (Contributed by AV, 22-May-2024.) |
β’ π» = (β β (2-aryF π) β¦ (π₯ β π, π¦ β π β¦ (ββ{β¨0, π₯β©, β¨1, π¦β©}))) β β’ (π β π β π»:(2-aryF π)β1-1β(π βm (π Γ π))) | ||
Theorem | 2arymaptfo 47640* | The mapping of binary (endo)functions is a function onto the set of binary operations. (Contributed by AV, 23-May-2024.) |
β’ π» = (β β (2-aryF π) β¦ (π₯ β π, π¦ β π β¦ (ββ{β¨0, π₯β©, β¨1, π¦β©}))) β β’ (π β π β π»:(2-aryF π)βontoβ(π βm (π Γ π))) | ||
Theorem | 2arymaptf1o 47641* | The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations. (Contributed by AV, 23-May-2024.) |
β’ π» = (β β (2-aryF π) β¦ (π₯ β π, π¦ β π β¦ (ββ{β¨0, π₯β©, β¨1, π¦β©}))) β β’ (π β π β π»:(2-aryF π)β1-1-ontoβ(π βm (π Γ π))) | ||
Theorem | 2aryenef 47642 | The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.) |
β’ (2-aryF π) β (π βm (π Γ π)) | ||
According to Wikipedia ("Ackermann function", 8-May-2024, https://en.wikipedia.org/wiki/Ackermann_function): "In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. ... One common version is the two-argument Ackermann-PΓ©ter function developed by RΓ³zsa PΓ©ter and Raphael Robinson. Its value grows very rapidly; for example, A(4,2) results in 2^65536-3 [see ackval42 47682)], an integer of 19,729 decimal digits." In the following, the Ackermann function is defined as iterated 1-ary function (also mentioned in Wikipedia), see df-ack 47646, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 47645. As an illustration, we have ((IterCompβπΉ)β3) = (πΉ β (πΉ β πΉ))) (see itcoval3 47651). The following recursive definition of the Ackermann function follows immediately from Definition df-ack 47646: ((Ackβ(π + 1))βπ) = (((IterCompβ(Ackβπ))β(π + 1))β1)). That Definition df-ack 47646 is equivalent to PΓ©ter's definition is proven by the following three theorems: ackval0val 47672: ((Ackβ0)βπ) = (π + 1); ackvalsuc0val 47673: ((Ackβ(π + 1))β0) = ((Ackβπ)β1); ackvalsucsucval 47674: ((Ackβ(π + 1))β(π + 1)) = ((Ackβπ)β((Ackβ(π + 1))βπ)). The initial values of the Ackermann function are calculated in the following four theorems: ackval0012 47675: π΄(0, 0) = 1, π΄(0, 1) = 2, π΄(0, 2) = 3; ackval1012 47676: π΄(1, 0) = 2, π΄(1, 1) = 3, π΄(1, 3) = 4; ackval2012 47677: π΄(2, 0) = 3, π΄(2, 1) = 5, π΄(2, 3) = 7; ackval3012 47678: π΄(3, 0) = 5, π΄(3, 1) = ;13, π΄(3, 3) = ;29. | ||
Syntax | citco 47643 | Extend the definition of a class to include iterated functions. |
class IterComp | ||
Syntax | cack 47644 | Extend the definition of a class to include the Ackermann function operator. |
class Ack | ||
Definition | df-itco 47645* | Define a function (recursively) that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.) |
β’ IterComp = (π β V β¦ seq0((π β V, π β V β¦ (π β π)), (π β β0 β¦ if(π = 0, ( I βΎ dom π), π)))) | ||
Definition | df-ack 47646* | Define the Ackermann function (recursively). (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.) |
β’ Ack = seq0((π β V, π β V β¦ (π β β0 β¦ (((IterCompβπ)β(π + 1))β1))), (π β β0 β¦ if(π = 0, (π β β0 β¦ (π + 1)), π))) | ||
Theorem | itcoval 47647* | The value of the function that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by AV, 2-May-2024.) |
β’ (πΉ β π β (IterCompβπΉ) = seq0((π β V, π β V β¦ (πΉ β π)), (π β β0 β¦ if(π = 0, ( I βΎ dom πΉ), πΉ)))) | ||
Theorem | itcoval0 47648 | A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.) |
β’ (πΉ β π β ((IterCompβπΉ)β0) = ( I βΎ dom πΉ)) | ||
Theorem | itcoval1 47649 | A function iterated once. (Contributed by AV, 2-May-2024.) |
β’ ((Rel πΉ β§ πΉ β π) β ((IterCompβπΉ)β1) = πΉ) | ||
Theorem | itcoval2 47650 | A function iterated twice. (Contributed by AV, 2-May-2024.) |
β’ ((Rel πΉ β§ πΉ β π) β ((IterCompβπΉ)β2) = (πΉ β πΉ)) | ||
Theorem | itcoval3 47651 | A function iterated three times. (Contributed by AV, 2-May-2024.) |
β’ ((Rel πΉ β§ πΉ β π) β ((IterCompβπΉ)β3) = (πΉ β (πΉ β πΉ))) | ||
Theorem | itcoval0mpt 47652* | A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.) |
β’ πΉ = (π β π΄ β¦ π΅) β β’ ((π΄ β π β§ βπ β π΄ π΅ β π) β ((IterCompβπΉ)β0) = (π β π΄ β¦ π)) | ||
Theorem | itcovalsuc 47653* | The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.) |
β’ ((πΉ β π β§ π β β0 β§ ((IterCompβπΉ)βπ) = πΊ) β ((IterCompβπΉ)β(π + 1)) = (πΊ(π β V, π β V β¦ (πΉ β π))πΉ)) | ||
Theorem | itcovalsucov 47654 | The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.) |
β’ ((πΉ β π β§ π β β0 β§ ((IterCompβπΉ)βπ) = πΊ) β ((IterCompβπΉ)β(π + 1)) = (πΉ β πΊ)) | ||
Theorem | itcovalendof 47655 | The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.) |
β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆπ΄) & β’ (π β π β β0) β β’ (π β ((IterCompβπΉ)βπ):π΄βΆπ΄) | ||
Theorem | itcovalpclem1 47656* | Lemma 1 for itcovalpc 47658: induction basis. (Contributed by AV, 4-May-2024.) |
β’ πΉ = (π β β0 β¦ (π + πΆ)) β β’ (πΆ β β0 β ((IterCompβπΉ)β0) = (π β β0 β¦ (π + (πΆ Β· 0)))) | ||
Theorem | itcovalpclem2 47657* | Lemma 2 for itcovalpc 47658: induction step. (Contributed by AV, 4-May-2024.) |
β’ πΉ = (π β β0 β¦ (π + πΆ)) β β’ ((π¦ β β0 β§ πΆ β β0) β (((IterCompβπΉ)βπ¦) = (π β β0 β¦ (π + (πΆ Β· π¦))) β ((IterCompβπΉ)β(π¦ + 1)) = (π β β0 β¦ (π + (πΆ Β· (π¦ + 1)))))) | ||
Theorem | itcovalpc 47658* | The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.) |
β’ πΉ = (π β β0 β¦ (π + πΆ)) β β’ ((πΌ β β0 β§ πΆ β β0) β ((IterCompβπΉ)βπΌ) = (π β β0 β¦ (π + (πΆ Β· πΌ)))) | ||
Theorem | itcovalt2lem2lem1 47659 | Lemma 1 for itcovalt2lem2 47662. (Contributed by AV, 6-May-2024.) |
β’ (((π β β β§ πΆ β β0) β§ π β β0) β (((π + πΆ) Β· π) β πΆ) β β0) | ||
Theorem | itcovalt2lem2lem2 47660 | Lemma 2 for itcovalt2lem2 47662. (Contributed by AV, 7-May-2024.) |
β’ (((π β β0 β§ πΆ β β0) β§ π β β0) β ((2 Β· (((π + πΆ) Β· (2βπ)) β πΆ)) + πΆ) = (((π + πΆ) Β· (2β(π + 1))) β πΆ)) | ||
Theorem | itcovalt2lem1 47661* | Lemma 1 for itcovalt2 47663: induction basis. (Contributed by AV, 5-May-2024.) |
β’ πΉ = (π β β0 β¦ ((2 Β· π) + πΆ)) β β’ (πΆ β β0 β ((IterCompβπΉ)β0) = (π β β0 β¦ (((π + πΆ) Β· (2β0)) β πΆ))) | ||
Theorem | itcovalt2lem2 47662* | Lemma 2 for itcovalt2 47663: induction step. (Contributed by AV, 7-May-2024.) |
β’ πΉ = (π β β0 β¦ ((2 Β· π) + πΆ)) β β’ ((π¦ β β0 β§ πΆ β β0) β (((IterCompβπΉ)βπ¦) = (π β β0 β¦ (((π + πΆ) Β· (2βπ¦)) β πΆ)) β ((IterCompβπΉ)β(π¦ + 1)) = (π β β0 β¦ (((π + πΆ) Β· (2β(π¦ + 1))) β πΆ)))) | ||
Theorem | itcovalt2 47663* | The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.) |
β’ πΉ = (π β β0 β¦ ((2 Β· π) + πΆ)) β β’ ((πΌ β β0 β§ πΆ β β0) β ((IterCompβπΉ)βπΌ) = (π β β0 β¦ (((π + πΆ) Β· (2βπΌ)) β πΆ))) | ||
Theorem | ackvalsuc1mpt 47664* | The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.) |
β’ (π β β0 β (Ackβ(π + 1)) = (π β β0 β¦ (((IterCompβ(Ackβπ))β(π + 1))β1))) | ||
Theorem | ackvalsuc1 47665 | The Ackermann function at a successor of the first argument and an arbitrary second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.) |
β’ ((π β β0 β§ π β β0) β ((Ackβ(π + 1))βπ) = (((IterCompβ(Ackβπ))β(π + 1))β1)) | ||
Theorem | ackval0 47666 | The Ackermann function at 0. (Contributed by AV, 2-May-2024.) |
β’ (Ackβ0) = (π β β0 β¦ (π + 1)) | ||
Theorem | ackval1 47667 | The Ackermann function at 1. (Contributed by AV, 4-May-2024.) |
β’ (Ackβ1) = (π β β0 β¦ (π + 2)) | ||
Theorem | ackval2 47668 | The Ackermann function at 2. (Contributed by AV, 4-May-2024.) |
β’ (Ackβ2) = (π β β0 β¦ ((2 Β· π) + 3)) | ||
Theorem | ackval3 47669 | The Ackermann function at 3. (Contributed by AV, 7-May-2024.) |
β’ (Ackβ3) = (π β β0 β¦ ((2β(π + 3)) β 3)) | ||
Theorem | ackendofnn0 47670 | The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.) |
β’ (π β β0 β (Ackβπ):β0βΆβ0) | ||
Theorem | ackfnnn0 47671 | The Ackermann function at any nonnegative integer is a function on the nonnegative integers. (Contributed by AV, 4-May-2024.) (Proof shortened by AV, 8-May-2024.) |
β’ (π β β0 β (Ackβπ) Fn β0) | ||
Theorem | ackval0val 47672 | The Ackermann function at 0 (for the first argument). This is the first equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
β’ (π β β0 β ((Ackβ0)βπ) = (π + 1)) | ||
Theorem | ackvalsuc0val 47673 | The Ackermann function at a successor (of the first argument). This is the second equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
β’ (π β β0 β ((Ackβ(π + 1))β0) = ((Ackβπ)β1)) | ||
Theorem | ackvalsucsucval 47674 | The Ackermann function at the successors. This is the third equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.) |
β’ ((π β β0 β§ π β β0) β ((Ackβ(π + 1))β(π + 1)) = ((Ackβπ)β((Ackβ(π + 1))βπ))) | ||
Theorem | ackval0012 47675 | The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.) |
β’ β¨((Ackβ0)β0), ((Ackβ0)β1), ((Ackβ0)β2)β© = β¨1, 2, 3β© | ||
Theorem | ackval1012 47676 | The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.) |
β’ β¨((Ackβ1)β0), ((Ackβ1)β1), ((Ackβ1)β2)β© = β¨2, 3, 4β© | ||
Theorem | ackval2012 47677 | The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.) |
β’ β¨((Ackβ2)β0), ((Ackβ2)β1), ((Ackβ2)β2)β© = β¨3, 5, 7β© | ||
Theorem | ackval3012 47678 | The Ackermann function at (3,0), (3,1), (3,2). (Contributed by AV, 7-May-2024.) |
β’ β¨((Ackβ3)β0), ((Ackβ3)β1), ((Ackβ3)β2)β© = β¨5, ;13, ;29β© | ||
Theorem | ackval40 47679 | The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ4)β0) = ;13 | ||
Theorem | ackval41a 47680 | The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ4)β1) = ((2β;16) β 3) | ||
Theorem | ackval41 47681 | The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ4)β1) = ;;;;65533 | ||
Theorem | ackval42 47682 | The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ4)β2) = ((2β;;;;65536) β 3) | ||
Theorem | ackval42a 47683 | The Ackermann function at (4,2), expressed with powers of 2. (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ4)β2) = ((2β(2β(2β(2β2)))) β 3) | ||
Theorem | ackval50 47684 | The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.) |
β’ ((Ackβ5)β0) = ;;;;65533 | ||
Theorem | fv1prop 47685 | The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.) |
β’ (π΄ β π β ({β¨1, π΄β©, β¨2, π΅β©}β1) = π΄) | ||
Theorem | fv2prop 47686 | The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.) |
β’ (π΅ β π β ({β¨1, π΄β©, β¨2, π΅β©}β2) = π΅) | ||
Theorem | submuladdmuld 47687 | Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π· β β) β β’ (π β (((π΄ β π΅) Β· πΆ) + (π΅ Β· π·)) = ((π΄ Β· πΆ) + (π΅ Β· (π· β πΆ)))) | ||
Theorem | affinecomb1 47688* | Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π΅ β πΆ) & β’ (π β πΈ β β) & β’ (π β πΉ β β) & β’ (π β πΊ β β) & β’ π = ((πΊ β πΉ) / (πΆ β π΅)) β β’ (π β (βπ‘ β β (π΄ = (((1 β π‘) Β· π΅) + (π‘ Β· πΆ)) β§ πΈ = (((1 β π‘) Β· πΉ) + (π‘ Β· πΊ))) β πΈ = ((π Β· (π΄ β π΅)) + πΉ))) | ||
Theorem | affinecomb2 47689* | Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π΅ β πΆ) & β’ (π β πΈ β β) & β’ (π β πΉ β β) & β’ (π β πΊ β β) β β’ (π β (βπ‘ β β (π΄ = (((1 β π‘) Β· π΅) + (π‘ Β· πΆ)) β§ πΈ = (((1 β π‘) Β· πΉ) + (π‘ Β· πΊ))) β ((πΆ β π΅) Β· πΈ) = (((πΊ β πΉ) Β· π΄) + ((πΉ Β· πΆ) β (π΅ Β· πΊ))))) | ||
Theorem | affineid 47690 | Identity of an affine combination. (Contributed by AV, 2-Feb-2023.) |
β’ (π β π΄ β β) & β’ (π β π β β) β β’ (π β (((1 β π) Β· π΄) + (π Β· π΄)) = π΄) | ||
Theorem | 1subrec1sub 47691 | Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.) |
β’ ((π΄ β β β§ π΄ β 1) β (1 β (1 / (1 β π΄))) = (π΄ / (π΄ β 1))) | ||
Theorem | resum2sqcl 47692 | The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.) |
β’ π = ((π΄β2) + (π΅β2)) β β’ ((π΄ β β β§ π΅ β β) β π β β) | ||
Theorem | resum2sqgt0 47693 | The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.) |
β’ π = ((π΄β2) + (π΅β2)) β β’ (((π΄ β β β§ π΄ β 0) β§ π΅ β β) β 0 < π) | ||
Theorem | resum2sqrp 47694 | The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.) |
β’ π = ((π΄β2) + (π΅β2)) β β’ (((π΄ β β β§ π΄ β 0) β§ π΅ β β) β π β β+) | ||
Theorem | resum2sqorgt0 47695 | The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.) |
β’ π = ((π΄β2) + (π΅β2)) β β’ ((π΄ β β β§ π΅ β β β§ (π΄ β 0 β¨ π΅ β 0)) β 0 < π) | ||
Theorem | reorelicc 47696 | Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (πΆ < π΄ β¨ πΆ β (π΄[,]π΅) β¨ π΅ < πΆ)) | ||
Theorem | rrx2pxel 47697 | The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.) |
β’ πΌ = {1, 2} & β’ π = (β βm πΌ) β β’ (π β π β (πβ1) β β) | ||
Theorem | rrx2pyel 47698 | The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.) |
β’ πΌ = {1, 2} & β’ π = (β βm πΌ) β β’ (π β π β (πβ2) β β) | ||
Theorem | prelrrx2 47699 | An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.) |
β’ πΌ = {1, 2} & β’ π = (β βm πΌ) β β’ ((π΄ β β β§ π΅ β β) β {β¨1, π΄β©, β¨2, π΅β©} β π) | ||
Theorem | prelrrx2b 47700 | An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2, determined by its coordinates. (Contributed by AV, 7-May-2023.) |
β’ πΌ = {1, 2} & β’ π = (β βm πΌ) β β’ (((π΄ β β β§ π΅ β β) β§ (π β β β§ π β β)) β ((π β π β§ (((πβ1) = π΄ β§ (πβ2) = π΅) β¨ ((πβ1) = π β§ (πβ2) = π))) β π β {{β¨1, π΄β©, β¨2, π΅β©}, {β¨1, πβ©, β¨2, πβ©}})) |
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