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Intuitionistic Logic Explorer Theorem List (p. 156 of 156) | < Previous Wrap > |
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Type | Label | Description |
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Statement | ||
Theorem | nconstwlpo 15501* |
Existence of a certain non-constant function from reals to integers
implies ![]() ![]() |
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Theorem | neapmkvlem 15502* | Lemma for neapmkv 15503. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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Theorem | neapmkv 15503* | 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.) |
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Theorem | neap0mkv 15504* | 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.) |
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Theorem | ltlenmkv 15505* |
If ![]() ![]() |
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Theorem | supfz 15506 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
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Theorem | inffz 15507 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
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Theorem | taupi 15508 |
Relationship between ![]() ![]() |
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Theorem | ax1hfs 15509 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) |
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Theorem | dftest 15510 |
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 |
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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 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
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 15511 |
Extend wff definition to include "all some" applied to a top-level
implication, which means ![]() ![]() ![]() ![]() |
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Syntax | walsc 15512 |
Extend wff definition to include "all some" applied to a class, which
means ![]() ![]() ![]() ![]() ![]() |
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Definition | df-alsi 15513 |
Define "all some" applied to a top-level implication, which means
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Definition | df-alsc 15514 |
Define "all some" applied to a class, which means ![]() ![]() ![]() ![]() ![]() |
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Theorem | alsconv 15515 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
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Theorem | alsi1d 15516 | 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.) |
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Theorem | alsi2d 15517 | 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.) |
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Theorem | alsc1d 15518 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
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Theorem | alsc2d 15519 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
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