P. 158 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15701-15728   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	redc0 15701*	Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
|- ( A. x e. RR A. y e. RR DECID x = y <-> A. z e. RR DECID z = 0 )
Theorem	reap0 15702*	Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. z e. RR DECID z # 0 )
Theorem	cndcap 15703*	Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. z e. CC A. w e. CC DECID z # w )
Theorem	dceqnconst 15704*	Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 15699 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
|- ( A. x e. RR DECID x = 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
Theorem	dcapnconst 15705*	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 15687 for more discussion of decidability of real number apartness. This is a weaker form of dceqnconst 15704 and in fact this theorem can be proved using dceqnconst 15704 as shown at dcapnconstALT 15706. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)
|- ( A. x e. RR DECID x # 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
Theorem	dcapnconstALT 15706*	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 15705 by means of dceqnconst 15704. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A. x e. RR DECID x # 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
Theorem	nconstwlpolem0 15707*	Lemma for nconstwlpo 15710. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
|- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   &    |- ( ph -> A. x e. NN ( G ` x ) = 0 )   =>    |- ( ph -> A = 0 )
Theorem	nconstwlpolemgt0 15708*	Lemma for nconstwlpo 15710. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
|- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   &    |- ( ph -> E. x e. NN ( G ` x ) = 1 )   =>    |- ( ph -> 0 < A )
Theorem	nconstwlpolem 15709*	Lemma for nconstwlpo 15710. (Contributed by Jim Kingdon, 23-Jul-2024.)
|- ( ph -> F : RR --> ZZ )   &    |- ( ph -> ( F ` 0 ) = 0 )   &    |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )   &    |- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   =>    |- ( ph -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )
Theorem	nconstwlpo 15710*	Existence of a certain non-constant function from reals to integers implies om e. WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
|- ( ph -> F : RR --> ZZ )   &    |- ( ph -> ( F ` 0 ) = 0 )   &    |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )   =>    |- ( ph -> om e. WOmni )
Theorem	neapmkvlem 15711*	Lemma for neapmkv 15712. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ( ph /\ A =/= 1 ) -> A # 1 )   =>    |- ( ph -> ( -. A. x e. NN ( F ` x ) = 1 -> E. x e. NN ( F ` x ) = 0 ) )
Theorem	neapmkv 15712*	If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> om e. Markov )
Theorem	neap0mkv 15713*	The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
|- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) <-> A. x e. RR ( x =/= 0 -> x # 0 ) )
Theorem	ltlenmkv 15714*	If < can be expressed as holding exactly when <_ holds and the values are not equal, then the analytic Markov's Principle applies. (To get the regular Markov's Principle, combine with neapmkv 15712). (Contributed by Jim Kingdon, 23-Feb-2025.)
|- ( A. x e. RR A. y e. RR ( x < y <-> ( x <_ y /\ y =/= x ) ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
13.3.8  Supremum and infimum
Theorem	supfz 15715	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
|- ( N e. ( ZZ>= ` M ) -> sup ( ( M ... N ) , ZZ , < ) = N )
Theorem	inffz 15716	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
|- ( N e. ( ZZ>= ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M )
13.3.9  Circle constant
Theorem	taupi 15717	Relationship between tau and pi. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
|- tau = ( 2 x. pi )
13.4  Mathbox for Mykola Mostovenko
Theorem	ax1hfs 15718	Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
|- ( ph -> ( ph /\ ph ) )
13.5  Mathbox for David A. Wheeler
13.5.1  Testable propositions
Theorem	dftest 15719	A proposition is testable iff its negative or double-negative is true. See Chapter 2 [Moschovakis] p. 2. We do not formally define testability with a new token, but instead use DECID -. before the formula in question. For example, DECID -. x = y corresponds to " x = y is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) For statements about testable propositions, search for the keyword "testable" in the comments of statements, for instance using the Metamath command "MM> SEARCH * "testable" / COMMENTS". (New usage is discouraged.)
|- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
13.5.2  Allsome quantifier These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like A. x ph -> ps do not imply that ph is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow A.! x ( ph -> ps ), and when restricted (applied to a class) we allow A.! x e. A ph. The first symbol after the setvar variable must always be e. if it is the form applied to a class, and since e. cannot begin a wff, it is unambiguous. The -> looks like it would be a problem because ph or ps might include implications, but any implication arrow -> within any wff must be surrounded by parentheses, so only the implication arrow of A.! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome.
Syntax	walsi 15720	Extend wff definition to include "all some" applied to a top-level implication, which means ps is true whenever ph is true, and there is at least least one x where ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff A.! x ( ph -> ps )
Syntax	walsc 15721	Extend wff definition to include "all some" applied to a class, which means ph is true for all x in A, and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff A.! x e. A ph
Definition	df-alsi 15722	Define "all some" applied to a top-level implication, which means ps is true whenever ph is true and there is at least one x where ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( A.! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
Definition	df-alsc 15723	Define "all some" applied to a class, which means ph is true for all x in A and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( A.! x e. A ph <-> ( A. x e. A ph /\ E. x x e. A ) )
Theorem	alsconv 15724	There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
|- ( A.! x ( x e. A -> ph ) <-> A.! x e. A ph )
Theorem	alsi1d 15725	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( ph -> A.! x ( ps -> ch ) )   =>    |- ( ph -> A. x ( ps -> ch ) )
Theorem	alsi2d 15726	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( ph -> A.! x ( ps -> ch ) )   =>    |- ( ph -> E. x ps )
Theorem	alsc1d 15727	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( ph -> A.! x e. A ps )   =>    |- ( ph -> A. x e. A ps )
Theorem	alsc2d 15728	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
|- ( ph -> A.! x e. A ps )   =>    |- ( ph -> E. x x e. A )

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