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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | blennn0e2 45401 | 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.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘𝑁) = ((#b‘(𝑁 / 2)) + 1)) | ||
Generalization of df-bits 15826. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 45421: if 𝐾 and 𝑁 are nonnegative integers, then ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)). | ||
Syntax | cdig 45402 | Extend class notation with the class of the digit extraction operation. |
class digit | ||
Definition | df-dig 45403* | Definition of an operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝑏. 𝑘 = − 1 corresponds to the first digit of the fractional part (for 𝑏 = 10 the first digit after the decimal point), 𝑘 = 0 corresponds to the last digit of the integer part (for 𝑏 = 10 the first digit before the decimal point). See also digit1 13653. 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.) |
⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏))) | ||
Theorem | digfval 45404* | Operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ (𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) | ||
Theorem | digval 45405 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) | ||
Theorem | digvalnn0 45406 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0) | ||
Theorem | nn0digval 45407 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵↑𝐾))) mod 𝐵)) | ||
Theorem | dignn0fr 45408 | The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)𝑁) = 0) | ||
Theorem | dignn0ldlem 45409 | Lemma for dignnld 45410. (Contributed by AV, 25-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾)) | ||
Theorem | dignnld 45410 | The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0) | ||
Theorem | dig2nn0ld 45411 | The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘(#b‘𝑁))) → (𝐾(digit‘2)𝑁) = 0) | ||
Theorem | dig2nn1st 45412 | The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.) |
⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = 1) | ||
Theorem | dig0 45413 | All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0) | ||
Theorem | digexp 45414 | The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵↑𝑁)) = if(𝐾 = 𝑁, 1, 0)) | ||
Theorem | dig1 45415 | All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0)) | ||
Theorem | 0dig1 45416 | The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.) |
⊢ (𝐵 ∈ (ℤ≥‘2) → (0(digit‘𝐵)1) = 1) | ||
Theorem | 0dig2pr01 45417 | The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.) |
⊢ (𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁) | ||
Theorem | dig2nn0 45418 | A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1}) | ||
Theorem | 0dig2nn0e 45419 | The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0) | ||
Theorem | 0dig2nn0o 45420 | The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 1) | ||
Theorem | dig2bits 45421 | The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁))) | ||
Theorem | dignn0flhalflem1 45422 | Lemma 1 for dignn0flhalf 45425. (Contributed by AV, 7-Jun-2012.) |
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁)))) | ||
Theorem | dignn0flhalflem2 45423 | Lemma 2 for dignn0flhalf 45425. (Contributed by AV, 7-Jun-2012.) |
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁)))) | ||
Theorem | dignn0ehalf 45424 | 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.) |
⊢ (((𝐴 / 2) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2))) | ||
Theorem | dignn0flhalf 45425 | 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.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2)))) | ||
Theorem | nn0sumshdiglemA 45426* | Lemma for nn0sumshdig 45430 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.) |
⊢ (((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglemB 45427* | Lemma for nn0sumshdig 45430 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.) |
⊢ (((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglem1 45428* | Lemma 1 for nn0sumshdig 45430 (induction step). (Contributed by AV, 7-Jun-2020.) |
⊢ (𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b‘𝑎) = 𝑦 → 𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglem2 45429* | Lemma 2 for nn0sumshdig 45430. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b‘𝑎) = 𝐿 → 𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘)))) | ||
Theorem | nn0sumshdig 45430* | A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 = Σ𝑘 ∈ (0..^(#b‘𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘))) | ||
Theorem | nn0mulfsum 45431* | Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding 𝑏 to itself 𝑎 times. (Contributed by AV, 17-May-2020.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵) | ||
Theorem | nn0mullong 45432* | Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e., the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15897. (Contributed by AV, 7-Jun-2020.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b‘𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵)) | ||
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: 𝑓:𝑈⟶𝑌 where where the domain 𝑈 has the form 𝑈 ⊆ ((...((𝑋‘1) × (𝑋‘2)) × ...) × (𝑋‘𝑛))." In the following, n-ary functions are defined as mappings (see df-map 8423) 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 45442 and mapsn 8475: (𝑋 ↑m {∅}) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 45452 and efmndbas 18107: (𝑋 ↑m 𝑋) - 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 18107): "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 45463 and compare with df-clintop 44855: (𝑋 ↑m (𝑋 × 𝑋)). Instead of using indexed arguments (represented by a mapping as described above), elements of Cartesian exponentiations (𝑈↑↑𝑁) (see df-finxp 35107) 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., (𝑈↑↑2o), instead of integers, e.g., (𝑈↑↑2), 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 45436 shows that both definitions are equivalent. | ||
Syntax | cnaryf 45433 | Extend the definition of a class to include the n-ary functions. |
class -aryF | ||
Definition | df-naryf 45434* | Define the n-ary (endo)functions. (Contributed by AV, 11-May-2024.) (Revised by TA and SN, 7-Jun-2024.) |
⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛)))) | ||
Theorem | naryfval 45435 | The set of the n-ary (endo)functions on a class 𝑋. (Contributed by AV, 13-May-2024.) |
⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) | ||
Theorem | naryfvalixp 45436* | The set of the n-ary (endo)functions on a class 𝑋 expressed with the notation of infinite Cartesian products. (Contributed by AV, 19-May-2024.) |
⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)) | ||
Theorem | naryfvalel 45437 | An n-ary (endo)function on a set 𝑋. (Contributed by AV, 14-May-2024.) |
⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) | ||
Theorem | naryrcl 45438 | Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.) |
⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) | ||
Theorem | naryfvalelfv 45439 | The value of an n-ary (endo)function on a set 𝑋 is an element of 𝑋. (Contributed by AV, 14-May-2024.) |
⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋) | ||
Theorem | naryfvalelwrdf 45440* | An n-ary (endo)function on a set 𝑋 expressed as a function over the set of words on 𝑋 of length 𝑛. (Contributed by AV, 4-Jun-2024.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:{𝑤 ∈ Word 𝑋 ∣ (♯‘𝑤) = 𝑁}⟶𝑋)) | ||
Theorem | 0aryfvalel 45441* | A nullary (endo)function on a set 𝑋 is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: (𝐹‘∅) = 𝐶 with 𝐶 ∈ 𝑋, see also 0aryfvalelfv 45442. Instead of (𝐹‘∅), nullary functions are usually written as 𝐹() in literature. (Contributed by AV, 15-May-2024.) |
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) | ||
Theorem | 0aryfvalelfv 45442* | The value of a nullary (endo)function on a set 𝑋. (Contributed by AV, 19-May-2024.) |
⊢ (𝐹 ∈ (0-aryF 𝑋) → ∃𝑥 ∈ 𝑋 (𝐹‘∅) = 𝑥) | ||
Theorem | 1aryfvalel 45443 | A unary (endo)function on a set 𝑋. (Contributed by AV, 15-May-2024.) |
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (1-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m {0})⟶𝑋)) | ||
Theorem | fv1arycl 45444 | Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.) |
⊢ ((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐺‘{〈0, 𝐴〉}) ∈ 𝑋) | ||
Theorem | 1arympt1 45445* | A unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.) |
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0))) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴:𝑋⟶𝑋) → 𝐹 ∈ (1-aryF 𝑋)) | ||
Theorem | 1arympt1fv 45446* | The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.) |
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0))) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{〈0, 𝐵〉}) = (𝐴‘𝐵)) | ||
Theorem | 1arymaptfv 45447* | The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) ⇒ ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) | ||
Theorem | 1arymaptf 45448* | The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑m 𝑋)) | ||
Theorem | 1arymaptf1 45449* | The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑m 𝑋)) | ||
Theorem | 1arymaptfo 45450* | The mapping of unary (endo)functions is a function onto the set of endofunctions. (Contributed by AV, 18-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–onto→(𝑋 ↑m 𝑋)) | ||
Theorem | 1arymaptf1o 45451* | The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions (Contributed by AV, 19-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1-onto→(𝑋 ↑m 𝑋)) | ||
Theorem | 1aryenef 45452 | The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.) |
⊢ (1-aryF 𝑋) ≈ (𝑋 ↑m 𝑋) | ||
Theorem | 1aryenefmnd 45453 | The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.) |
⊢ (1-aryF 𝑋) ≈ (Base‘(EndoFMnd‘𝑋)) | ||
Theorem | 2aryfvalel 45454 | A binary (endo)function on a set 𝑋. (Contributed by AV, 20-May-2024.) |
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (2-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m {0, 1})⟶𝑋)) | ||
Theorem | fv2arycl 45455 | Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.) |
⊢ ((𝐺 ∈ (2-aryF 𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐺‘{〈0, 𝐴〉, 〈1, 𝐵〉}) ∈ 𝑋) | ||
Theorem | 2arympt 45456* | A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.) |
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1))) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑂:(𝑋 × 𝑋)⟶𝑋) → 𝐹 ∈ (2-aryF 𝑋)) | ||
Theorem | 2arymptfv 45457* | The value of a binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.) |
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1))) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{〈0, 𝐴〉, 〈1, 𝐵〉}) = (𝐴𝑂𝐵)) | ||
Theorem | 2arymaptfv 45458* | The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) ⇒ ⊢ (𝐹 ∈ (2-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) | ||
Theorem | 2arymaptf 45459* | The mapping of binary (endo)functions is a function into the set of binary operations. (Contributed by AV, 21-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋))) | ||
Theorem | 2arymaptf1 45460* | The mapping of binary (endo)functions is a one-to-one function into the set of binary operations. (Contributed by AV, 22-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1→(𝑋 ↑m (𝑋 × 𝑋))) | ||
Theorem | 2arymaptfo 45461* | The mapping of binary (endo)functions is a function onto the set of binary operations. (Contributed by AV, 23-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–onto→(𝑋 ↑m (𝑋 × 𝑋))) | ||
Theorem | 2arymaptf1o 45462* | The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations (Contributed by AV, 23-May-2024.) |
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋))) | ||
Theorem | 2aryenef 45463 | The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.) |
⊢ (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋)) | ||
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-Péter function developed by Rózsa Péter and Raphael Robinson. Its value grows very rapidly; for example, A(4,2) results in 2^65536-3 [see ackval42 45503)], 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 45467, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 45466. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹))) (see itcoval3 45472). The following recursive definition of the Ackermann function follows immediately from Definition df-ack 45467: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)). That Definition df-ack 45467 is equivalent to Péter's definition is proven by the following three theorems: ackval0val 45493: ((Ack‘0)‘𝑀) = (𝑀 + 1); ackvalsuc0val 45494: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1); ackvalsucsucval 45495: ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)). The initial values of the Ackermann function are calculated in the following four theorems: ackval0012 45496: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3; ackval1012 45497: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4; ackval2012 45498: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7; ackval3012 45499: 𝐴(3, 0) = 5, 𝐴(3, 1) = ;13, 𝐴(3, 3) = ;29. | ||
Syntax | citco 45464 | Extend the definition of a class to include iterated functions. |
class IterComp | ||
Syntax | cack 45465 | Extend the definition of a class to include the Ackermann function operator. |
class Ack | ||
Definition | df-itco 45466* | 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.) |
⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓)))) | ||
Definition | df-ack 45467* | Define the Ackermann function (recursively). (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.) |
⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))) | ||
Theorem | itcoval 45468* | 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.) |
⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))) | ||
Theorem | itcoval0 45469 | A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.) |
⊢ (𝐹 ∈ 𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹)) | ||
Theorem | itcoval1 45470 | A function iterated once. (Contributed by AV, 2-May-2024.) |
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹) | ||
Theorem | itcoval2 45471 | A function iterated twice. (Contributed by AV, 2-May-2024.) |
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘2) = (𝐹 ∘ 𝐹)) | ||
Theorem | itcoval3 45472 | A function iterated three times. (Contributed by AV, 2-May-2024.) |
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹))) | ||
Theorem | itcoval0mpt 45473* | A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.) |
⊢ 𝐹 = (𝑛 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ 𝑊) → ((IterComp‘𝐹)‘0) = (𝑛 ∈ 𝐴 ↦ 𝑛)) | ||
Theorem | itcovalsuc 45474* | 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.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹)) | ||
Theorem | itcovalsucov 45475 | 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.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹 ∘ 𝐺)) | ||
Theorem | itcovalendof 45476 | The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) | ||
Theorem | itcovalpclem1 45477* | Lemma 1 for itcovalpc 45479: induction basis. (Contributed by AV, 4-May-2024.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ⇒ ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0)))) | ||
Theorem | itcovalpclem2 45478* | Lemma 2 for itcovalpc 45479: induction step. (Contributed by AV, 4-May-2024.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ⇒ ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))) | ||
Theorem | itcovalpc 45479* | 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.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ⇒ ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼)))) | ||
Theorem | itcovalt2lem2lem1 45480 | Lemma 1 for itcovalt2lem2 45483. (Contributed by AV, 6-May-2024.) |
⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0) | ||
Theorem | itcovalt2lem2lem2 45481 | Lemma 2 for itcovalt2lem2 45483. (Contributed by AV, 7-May-2024.) |
⊢ (((𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶)) | ||
Theorem | itcovalt2lem1 45482* | Lemma 1 for itcovalt2 45484: induction basis. (Contributed by AV, 5-May-2024.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶)) ⇒ ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶))) | ||
Theorem | itcovalt2lem2 45483* | Lemma 2 for itcovalt2 45484: induction step. (Contributed by AV, 7-May-2024.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶)) ⇒ ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))) | ||
Theorem | itcovalt2 45484* | 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.) |
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶)) ⇒ ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))) | ||
Theorem | ackvalsuc1mpt 45485* | 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.) |
⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))) | ||
Theorem | ackvalsuc1 45486 | 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.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)) | ||
Theorem | ackval0 45487 | The Ackermann function at 0. (Contributed by AV, 2-May-2024.) |
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) | ||
Theorem | ackval1 45488 | The Ackermann function at 1. (Contributed by AV, 4-May-2024.) |
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) | ||
Theorem | ackval2 45489 | The Ackermann function at 2. (Contributed by AV, 4-May-2024.) |
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) | ||
Theorem | ackval3 45490 | The Ackermann function at 3. (Contributed by AV, 7-May-2024.) |
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3)) | ||
Theorem | ackendofnn0 45491 | The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.) |
⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0) | ||
Theorem | ackfnnn0 45492 | 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.) |
⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0) | ||
Theorem | ackval0val 45493 | The Ackermann function at 0 (for the first argument). This is the first equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
⊢ (𝑀 ∈ ℕ0 → ((Ack‘0)‘𝑀) = (𝑀 + 1)) | ||
Theorem | ackvalsuc0val 45494 | The Ackermann function at a successor (of the first argument). This is the second equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
⊢ (𝑀 ∈ ℕ0 → ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1)) | ||
Theorem | ackvalsucsucval 45495 | The Ackermann function at the successors. This is the third equation of Péter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁))) | ||
Theorem | ackval0012 45496 | The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.) |
⊢ 〈((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)〉 = 〈1, 2, 3〉 | ||
Theorem | ackval1012 45497 | The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.) |
⊢ 〈((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)〉 = 〈2, 3, 4〉 | ||
Theorem | ackval2012 45498 | The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.) |
⊢ 〈((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)〉 = 〈3, 5, 7〉 | ||
Theorem | ackval3012 45499 | The Ackermann function at (3,0), (3,1), (3,2). (Contributed by AV, 7-May-2024.) |
⊢ 〈((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)〉 = 〈5, ;13, ;29〉 | ||
Theorem | ackval40 45500 | The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.) |
⊢ ((Ack‘4)‘0) = ;13 |
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