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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-smo 5901* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | dfsmo2 5902* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Theorem | issmo 5903* | Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | issmo2 5904* | Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smoeq 5905 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smodm 5906 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smores 5907 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Theorem | smores3 5908 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Theorem | smores2 5909 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Theorem | smodm2 5910 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smofvon2dm 5911 | The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | iordsmo 5912 | The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smo0 5913 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smofvon 5914 | If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smoel 5915 | If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiun 5916* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiso 5917 | If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.) |
Theorem | smoel2 5918 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Syntax | crecs 5919 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 5920* |
Define a function recs on , the class of ordinal
numbers, by transfinite recursion given a rule which sets the next
value given all values so far. See df-irdg 5957 for more details on why
this definition is desirable. Unlike df-irdg 5957 which restricts the
update rule to use only the previous value, this version allows the
update rule to use all previous values, which is why it is
described
as "strong", although it is actually more primitive. See tfri1d 5949 and
tfri2d 5950 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 5921 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 5922 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 5923* | A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlem3ag 5924* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3a 5925* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3 5926* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3-2 5927* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.) |
Theorem | tfrlem3-2d 5928* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 5929* | Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem5 5930* | Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | recsfval 5931* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 5932* | Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem7 5933* | Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfrlem8 5934* | Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
recs | ||
Theorem | tfrlem9 5935* | Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.) |
recs recs recs | ||
Theorem | tfr2a 5936 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0 5937 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 5938* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 5939* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 5940* | Each element of is an acceptable function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 5941* | The union of is defined on all ordinals. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 5942* | The union of is a function defined on . Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 5943* | The set exists. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 5944* | The union of satisfies the recursion rule (lemma for tfrlemi1 5946). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 5945* | Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 5946* |
We can define an acceptable function on any ordinal.
As with many of the transfinite recursion theorems, we have a hypothesis that states that is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi14d 5947* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 5948* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 5949* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2d 5950* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri1 5951* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2 5952* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri3 5953* | Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfrex 5954* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfrfun 5955 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
recs | ||
Syntax | crdg 5956 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 5957* |
Define a recursive definition generator on (the class of ordinal
numbers) with characteristic function and initial value .
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our operation (especially when df-recs 5920
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple. In classical logic it would be easier to divide this definition
into cases based on whether the domain of is zero, a successor, or
a limit ordinal. Cases do not (in general) work that way in
intuitionistic logic, so instead we choose a definition which takes the
union of all the results of the characteristic function for ordinals in
the domain of .
This means that this definition has the expected
properties for increasing and continuous ordinal functions, which
include ordinal addition and multiplication.
For finite recursion we also define df-frec 5978 and for suitable characteristic functions df-frec 5978 yields the same result as restricted to , as seen at frecrdg 5992. Note: We introduce with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.) |
recs | ||
Theorem | rdgeq1 5958 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 5959 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 5960 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 5961* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 5962* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 5963* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 5964 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 5965 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 5966 | The recursive definition generator is a function on ordinal numbers. The condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 5973; in cases like df-oadd 6005 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 5967* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 5968* | Value of the recursive definition generator. Lemma for rdgival 5969 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 5969* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 5970 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 5971* |
One way of describing the value of the recursive definition generator at
a successor. There is no condition on the characteristic function
other than
. Given that, the resulting expression
encompasses both the expected successor term
but also
terms that correspond to
the initial value and to limit ordinals
.
If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5972. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 5972* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6044 and omsuc 6051. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 5973* | Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdg0 5974 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 5975 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 5976 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 5977 | Extend class notation with the fnite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 5978* |
Define a recursive definition generator on (the class of finite
ordinals) with characteristic function and initial value .
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our frec operation (especially when df-recs 5920
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple; see frec0g 5983 and frecsuc 5991.
Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4327. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 5992, this definition and df-irdg 5957 restricted to produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.) |
frec recs | ||
Theorem | freceq1 5979 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 5980 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 5981 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | nffrec 5982 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 5983 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 5984* | The class abstraction from df-frec 5978 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frectfr 5985* |
Lemma to connect transfinite recursion theorems with finite recursion.
That is, given the conditions
and on
frec , we
want to be able to apply tfri1d 5949 or tfri2d 5950,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 5986* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | frecsuclem1 5987* | Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 13-Aug-2019.) |
frec recs | ||
Theorem | frecsuclemdm 5988* | Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs | ||
Theorem | frecsuclem2 5989* | Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs frec | ||
Theorem | frecsuclem3 5990* | Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecsuc 5991* | The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecrdg 5992* |
Transfinite recursion restricted to omega.
Given a suitable characteristic function, df-frec 5978 produces the same results as df-irdg 5957 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
frec | ||
Theorem | freccl 5993* | Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.) |
frec | ||
Syntax | c1o 5994 | Extend the definition of a class to include the ordinal number 1. |
Syntax | c2o 5995 | Extend the definition of a class to include the ordinal number 2. |
Syntax | c3o 5996 | Extend the definition of a class to include the ordinal number 3. |
Syntax | c4o 5997 | Extend the definition of a class to include the ordinal number 4. |
Syntax | coa 5998 | Extend the definition of a class to include the ordinal addition operation. |
Syntax | comu 5999 | Extend the definition of a class to include the ordinal multiplication operation. |
Syntax | coei 6000 | Extend the definition of a class to include the ordinal exponentiation operation. |
↑_{𝑜} |
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