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Intuitionistic Logic Proof Explorer

Intuitionistic Logic (Wikipedia [accessed 19-Jul-2015], Stanford Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be thought of as a constructive logic in which we must build and exhibit concrete examples of objects before we can accept their existence. Unproved statements in intuitionistic logic are not given an intermediate truth value, instead, they remain of unknown truth value until they are either proved or disproved. Intuitionist logic can also be thought of as a weakening of classical logic such that the law of excluded middle (LEM), (φ ∨ ¬ φ), doesn't always hold. Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if we don't have a proof of either one. There is also no rule for double negation elimination. Brouwer observed in 1908 that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.

Contents of this page Overview of intuitionistic logic Overview of this work The axioms Some theorems How to intuitionize classical proofs Metamath Proof Explorer cross reference Bibliography | Related pages Table of Contents and Theorem List Most Recent Proofs (this mirror) (latest) Bibliographic Cross-Reference Definition List ASCII Equivalents for Text-Only Browsers Metamath database iset.mm (ASCII file) External links GitHub repository [accessed 06-Jan-2018]

Overview of intuitionistic logic

(Placeholder for future use)

Overview of this work

(By Gérard Lang, 7-May-2018)

Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a development of "set.mm" whose eventual aim is to show how many of the theorems of set theory and mathematics that can be derived from classical first order logic can also be derived from a weaker system called "intuitionistic logic." To achieve this task, iset.mm adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of set.mm.

Among these new axioms, the first six ( ~ ax-ia1 , ~ ax-ia2 , ~ ax-ia3 , ~ ax-io , ~ ax-in1 , and ~ ax-in2 ), when added to ~ ax-1 , ~ ax-2 , and ~ ax-mp , allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) (which is ax-3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax-3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equivalence of a proposition with its double negation, cannot be derived in intuitionistic propositional calculus. Glivenko showed that a proposition φ is a theorem of classical propositional calculus if and only if ¬¬φ is a theorem of intuitionistic propositional calculus.

The next 4 new axioms ( ~ ax-ial , ~ ax-i5r , ~ ax-ie1 , and ~ ax-ie2 ) together with the set.mm axioms ~ ax-4 , ~ ax-5 , ~ ax-7 , and ~ ax-gen do not mention equality or distinct variables.

The ~ ax-i9 axiom is just a slight variation of the classical ~ ax-9 . The classical axiom ~ ax-12 is strengthened into first ~ ax-i12 and then ~ ax-bndl (two results which would be fairly readily equivalent to ~ ax-12 classically but which do not follow from ~ ax-12 , at least not in an obvious way, in intuitionistic logic). The substitution of ~ ax-i9 , ~ ax-i12 , and ~ ax-bndl for ~ ax-9 and ~ ax-12 and the inclusion of ~ ax-8 , ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-14 , and ~ ax-17 allow for the development of the intuitionistic predicate calculus.

Each of the new axioms is a theorem of classical first order logic with equality. But some axioms of classical first order logic with equality, like ax-3, cannot be derived in the intuitionistic predicate calculus.

One of the major interests of the intuitionistic predicate calculus is that its use can be considered as a realization of the program of the constructivist philosophical view of mathematics.

The axioms

As with the classical axioms we have propositional logic and predicate logic.

The axioms of intuitionistic propositional logic consist of some of the axioms from classical propositional logic, plus additional axioms for the operation of the 'and', 'or' and 'not' connectives.

Axioms of intuitionistic propositional calculus
Axiom Simp | ax-1 | ⊢ (φ → (ψ → φ))
Axiom Frege | ax-2 | ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))
Rule of Modus Ponens | ax-mp | ` |- ph ` & ` |- ph -> ps ` ⇒ ` |- ps `
Left 'and' elimination | ax-ia1 | ` |- ( ( ph /\ ps ) -> ph ) `
Right 'and' elimination | ax-ia2 | ⊢ ((φ ∧ ψ) → ψ)
'And' introduction | ax-ia3 | ⊢ (φ → (ψ → (φ ∧ ψ)))
Definition of 'or' | ax-io | ⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ)))
'Not' introduction | ax-in1 | ⊢ ((φ → ¬ φ) → ¬ φ)
'Not' elimination | ax-in2 | ⊢ (¬ φ → (φ → ψ))

Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φ ∨ ψ is not equivalent to ¬ φ → ψ, nor is φ → ψ equivalent to ¬ φ ∨ ψ, nor is φ ∧ ψ equivalent to ¬ (φ → ¬ ψ).

The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if φ implies a contradiction (such as its own negation), then one can conclude ¬ φ. By contrast, assuming ¬ φ and then deriving a contradiction only serves to prove ¬ ¬ φ, which in intuitionistic logic is not the same as φ.

The biconditional can be defined as the conjunction of two implications, as in dfbi2 and df-bi.

Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃ cannot be defined in terms of ∀. As in classical logic, we also add = for equality (which is key to how we handle substitution in metamath) and ∈ (which for current purposes can just be thought of as an arbitrary predicate, but which will later come to mean set membership).

Our axioms are based on the classical set.mm predicate logic axioms, but adapted for intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing some aspects of how we handle negations.

Axioms of intuitionistic predicate logic
Axiom of Specialization | ax-4 | ⊢ (∀xφ → φ)
Axiom of Quantified Implication | ax-5 | ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
The converse of ax-5o | ax-i5r | ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ))
Axiom of Quantifier Commutation | ax-7 | ⊢ (∀x∀yφ → ∀y∀xφ)
Rule of Generalization | ax-gen | ⊢ φ => ⊢ ∀xφ
x is bound in ∀xφ | ax-ial | ⊢ (∀xφ → ∀x∀xφ)
x is bound in ∃xφ | ax-ie1 | ⊢ (∃xφ → ∀x∃xφ)
Define existential quantification | ax-ie2 | ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))
Axiom of Equality | ax-8 | ⊢ (x = y → (x = z → y = z))
Axiom of Existence | ax-i9 | ⊢ ∃x x = y
Axiom of Quantifier Substitution | ax-10 | ⊢ (∀x x = y → ∀y y = x)
Axiom of Variable Substitution | ax-11 | ⊢ (x = y → (∀yφ → ∀x(x = y → φ)))
Axiom of Quantifier Introduction | ax-i12 | ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))
Axiom of Bundling | ax-bndl | ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y)))
Left Membership Equality | ax-13 | ⊢ (x = y → (x ∈ z → y ∈ z))
Right Membership Equality | ax-14 | ⊢ (x = y → (z ∈ x → z ∈ y))
Distinctness | ax-17 | ⊢ (φ → ∀xφ), where x does not occur in φ

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. These axioms were developed in response to Russell's Paradox, a discovery that revolutionized the foundations of mathematics and logic.

In the IZF axioms that follow, all set variables are assumed to be distinct. If you click on their links you will see the explicit distinct variable conditions.

Intuitionistic Zermelo-Fraenkel Set Theory (IZF)
Axiom of Extensionality | ax-ext | ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)
Axiom of Collection | ax-coll | ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ)
Axiom of Separation | ax-sep | ⊢ ∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ))
Axiom of Power Sets | ax-pow | ⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y)
Axiom of Pairing | ax-pr | ⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z)
Axiom of Union | ax-un | ⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y)
Axiom of Set Induction | ax-setind | ⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ)
Axiom of Infinity | ax-iinf | ⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))

We develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.

A Theorem Sampler

From a psychological point of view, learning constructive mathematics is agonizing, for it requires one to first unlearn certain deeply ingrained intuitions and habits acquired during classical mathematical training.
—Andrej Bauer

Listed here are some examples of starting points for your journey through the maze. Some are stated just as they would be in a non-constructive context; others are here to highlight areas which look different constructively. You should study some simple proofs from propositional calculus until you get the hang of it. Then try some predicate calculus and finally set theory.

The Theorem List shows the complete set of theorems in the database. You may also find the Bibliographic Cross-Reference useful.

Propositional calculus Law of identity Praeclarum theorema Contraposition introduction Double negation introduction Triple negation Definition of exclusive or Negation and the false constant Predicate calculus Existential and universal quantifier swap Existentially quantified implication x = x Definition of proper substitution Double existential uniqueness Set theory Commutative law for union A basic relationship between class and wff variables Two ways of saying "is a set" The ZF axiom of foundation implies excluded middle Russell's paradox Ordinal trichotomy implies excluded middle Mathematical (finite) induction Peano's postulates for arithmetic: 1 2 3 4 5 Two natural numbers are either equal or not equal (a special case of the law of the excluded middle which we can prove). A natural number is either zero or a successor The axiom of choice implies excluded middle Burali-Forti paradox Transfinite induction Closure law for ordinal addition Real and complex numbers Properties of apartness: 1 2 3 4

How to intuitionize classical proofs

For the student or master of classical mathematics, constructive mathematics can be baffling. One can get over some of the intial hurdles of understanding how constructive mathematics works and why it might be interesting by reading [Bauer] but that work does little to explain in concrete terms how to write proofs in intuitionistic logic. Fortunately, metamath helps with one of the biggest hurdles: noticing when one is even using the law of the excluded middle or the axiom of choice. But suppose you have a classical proof from the Metamath Proof Explorer and it fails to verify when you copy it over to the Intuitionistic Logic Explorer. What then? Here are some rules of thumb in converting classical proofs to intuitionistic ones.

- If you see case elimination ( pm2.61 or its variants) you'll probably end up with two theorems for the two cases. In particular, if the cases were ` A e. _V ` and ` -. A e. _V ` you probably just care about the ` A e. _V ` case. On the other hand, if the proposition being eliminated is decidable (for example due to ~ nndceq , ~ zdceq , ~ zdcle , ~ zdclt , ~ eluzdc , or ~ fzdcel ), then case elimination will work using theorems such as ~ df-dc and ~ mpjaodan .
- Non-empty almost always needs to be changed to inhabited (those terms are defined at ~ n0rf ).
- If the original proof relied on propositional/predicate logic which isn't a theorem of intuitionistic logic, maybe there is a way of expressing the logic more directly. This is perhaps the hardest one to put in cookbook form: you might need to try some things and see if anything works.
- If the original proof relied on df-ex so that it could prove a theorem for ` A. ` and then get ` E. ` for free (or vice versa), instead go look at the original proof and try to come up the analogues to each step for the other quantifier (for example, ~ cbvrexcsf , ~ sbcrext , ~ rexxpf ). Similarly, if you have a theorem for ` <_ ` and are trying to prove the corresponding theorem for ` < ` you'll probably need to use analogous steps rather than negation (examples: ~ leaddsub , ~ ltsub1 , ~ ltsub2 ).
- If you are dealing with a definition, try to find the best constructive definition from the literature ([HoTT] book, Stanford Encyclopedia of Philosophy, [Bauer], etc). Once you pick a definition, that'll affect the proofs which rely on that definition.
- If there is case elimination on whether variables are distinct, most of the time you just need the variant with distinct variables. Sometimes you can then remove the constraint with a temporary variable (e.g. the various sbco2* variants, ~ nfralya , ~ r19.3rm ).
- Sometimes one direction of a biconditional holds, or subset holds instead of equality. You might be able to keep the easy direction and worry about the other one later.
- If there is case elimination sometimes only one of the two cases is possible. For example, in ~ mosubopt the rest of the formula being proved constrains which case matters.
- If you need an additional condition (for example, because the original proof used case elimination) and you are proving a biconditional, consider whether both sides of the biconditional imply the condition. If so, you'll be able to prove the biconditional with that condition as an antecedent, and then use ~ pm5.21nii or one of its variants to remove the antecedent (example: ~ elxp4 ).
- If your proof relies on dveeq2 try ~ dveeq2or and likewise for the other things downstream of ~ ax-i12 or ~ ax-bndl .
- If you have a disjunction, be reluctant to turn it into an implication using ~ ord and the like. Instead, show that each disjunct implies what you are trying to prove and use ~ jaoi to join those two implications into something which can hook up to the disjunction.
- Disjunctive syllogism holds in intuitionistic logic and we state it a few ways (for example ~ orel1 and ~ ecased ) but we don't have a wide variety of convenience theorems. Unless we add those, you'll use ~ ord or something similar followed by ~ mpd or something similar. This may add a few steps but they are straightforward ones.
- If your proof is doing tricky things perhaps in the interest of shortness, try just expanding the definitions and applying logic in a straightforward way. See if this gets you a working (although perhaps longer) proof.
- If your proof relies on a biconditional in set.mm which isn't in iset.mm, see if one direction is in iset.mm and see which direction your proof is using. For example ~ 19.35-1 or ~ exnalim .
- If you are doing things with inhabited classes (beyond just applying existing inhabited class theorems), you may be able to dig up some predicate logic you haven't used in a while (e.g. ~ raaan ).
- Consider the possibility of giving up. Some things just won't have intuitionistic proofs. The more it looks like excluded middle or other non-intuitionistic statements, the more likely you are dealing with one of these. But it can be hard to have good intuition about this. In some cases it may be possible to ask "can I use this statement to prove ` ph \/ -. ph ` for an arbitrary proposition" (see ~ ordtriexmid for example ), but this is not always an easy technique to apply.
- Switching between ~ 2th and ~ 2false might help (e.g. ~ dfnul2 , ~ dfnul3 , ~ rab0 ).
- In many cases statements which are equivalent in classical logic become several non-equivalent statements (e.g. exclusive or, ordinals, non-empty versus inhabited, apartness versus negated equality). This is usually a good place to look for a literature reference, but don't be afraid to change the statement being proved to "what you really meant is X" as appropriate.
- If a statement has multiple equivalences in set.mm (e.g. mo2 and mo3 , or dffun2 and dffun3 ) and only some of them in iset.mm, sometimes a pretty similar proof will work (that is, which one to use in the original proof may have been a fairly arbitrary choice).
- A number of theorems related to functions (especially ovex and fvex ) in set.mm perform case elimination based on whether we are evaluating the function within its domain or outside it. The most straightforward solution is to use ~ fnovex or ~ funfvex which only work within the domain. Using these may involve rearranging logic, for example by changing ~ rexlimivw to ~ rexlimdva (example: ~ ovelrn or indeed most uses of ~ fnovex and ~ funfvex ). If a function value is inhabited, we know we are evaluating it within its domain by ~ relelfvdm .
- With excluded middle, ` (/) e. A ` and ` A =/= (/) ` are equivalent (where ` A ` is an ordinal). In such theorems, ` (/) e. A ` is generally the more interesting condition constructively.
- Reverse closure in set.mm uses excluded middle (for example ovrcl or ndmfvrcl ). The most general way to handle this is to add more conditions that we are evaluating operations within their domains (for example set.mm's addasspi versus iset.mm's ~ addasspig in which conditions such as ` A e. N. ` are added, or set.mm's ltbtwnnq versus iset.mm's ~ ltbtwnnqq , in which ` E. x ` is changed to ` E. x e. Q. ` ). If the result of applying a function is inhabited, then we know we applied it within its domain - that is ~ relelfvdm or ~ elmpt2cl may be useful. (These thoughts apply to operations - in at least one place, ~ dvdszrcl , set.mm uses the term "reverse closure" for binary relations - but this is a lot more like ~ brel than like the reverse closure theorems described above).
- With excluded middle not equal (` =/= `) and apart (` # `) are equivalent. When working with real and complex numbers, apartness is almost always what you want. See ~ df-ap for more on apartness.
- Exclusive or ( ` ph \/_ ps ` ) is equivalent to ` ph <-> -. ps ` given excluded middle but we just have one direction ( ~ xorbin ). Consider intuitionizing ` ph <-> -. ps ` as ` ph \/_ ps ` (example: ~ rpnegap ).
- If you get stuck, ask! (for example in a GitHub issue or on the mailing list). We have a number of contributors who have experience in constructive mathematics in general, or iset.mm in particular, and one of the best things about doing/learning mathematics in metamath is the collaborative nature of how we develop it.

Metamath Proof Explorer cross reference

This is a list of theorems from the Metamath Proof Explorer (which assumes the law of the excluded middle throughout) which we do not have in the Intuitionistic Logic Explorer (generally because they are not provable without the law of the excluded middle, although some could be proved but aren't for a variety of reasons), together with the closest replacements.

set.mm | iset.mm | notes
--- | --- | ---
ax-3 , con4d , con34b , necon4bd | ~ con3 | The form of contraposition which removes negation does not hold in intutionistic logic.
pm2.18 | ~ pm2.01 | See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
pm2.18d , pm2.18i | ~ pm2.01d | See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
notnotrd , notnotri , notnotr , notnotb | ~ notnot | Double negation introduction holds but not double negation elimination.
mt3d | ~ mtod
nsyl2 | ~ nsyl
mt4d | ~ mt2d
pm2.61 , pm2.61d , pm2.61d1 , pm2.61d2 , pm2.61i , pm2.61ii , pm2.61nii , pm2.61iii , pm2.61ian , pm2.61dan , pm2.61ddan , pm2.61dda , pm2.61ine , pm2.61ne , pm2.61dne , pm2.61dane , pm2.61da2ne , pm2.61da3ne , pm2.61iine | none | If the proposition being eliminated is decidable (for example due to ~ nndceq , ~ zdceq , ~ zdcle , ~ zdclt , ~ eluzdc , or ~ fzdcel ), then case elimination will work using theorems such as ~ exmiddc and ~ mpjaodan
dfbi1 , dfbi3 | ~ df-bi , ~ dfbi2
impcon4bid, con4bid, notbi, con1bii, con4bii, con2bii | ~ con3 , ~ condc
xor3 , nbbn | ~ xorbin , ~ xornbi , ~ xor3dc , ~ nbbndc
biass | ~ biassdc
df-or , pm4.64 , pm2.54 , orri , orrd | ~ pm2.53 , ~ ori , ~ ord , ~ dfordc
imor , imori | ~ imorr , ~ imorri , ~ imordc
pm4.63 | ~ pm3.2im
iman | ~ imanim
annim | ~ annimim
oran , pm4.57 | ~ oranim , ~ orandc
ianor | ~ pm3.14 , ~ ianordc
pm4.14 | ~ pm4.14dc , ~ pm3.37
pm5.17 | ~ xorbin | The combination of ~ df-xor and ~ xorbin is the forward direction of pm5.17
biluk | ~ bilukdc
ecase | none | This is a form of case elimination. |
ecase3d | none | This is a form of case elimination. |
dedlem0b | ~ dedlemb
pm4.42 | ~ pm4.42r
3ianor | ~ 3ianorr
df-nan and other theorems using NAND (Sheffer stroke) notation | none | A quick glance at the internet shows this mostly being used in the presence of excluded middle; in any case it is not currently present in iset.mm
df-xor | ~ df-xor | Although the definition of exclusive or is called df-xor in both set.mm and iset.mm (at least currently), the definitions are not equivalent (in the absence of excluded middle).
xnor | none | The set.mm proof uses theorems not in iset.mm.
xorass | none | The set.mm proof uses theorems not in iset.mm.
xor2 | ~ xoranor , ~ xor2dc
xornan | ~ xor2dc
xornan2 | none | See discussion under df-nan
xorneg2 , xorneg1 , xorneg | none | The set.mm proofs use theorems not in iset.mm.
xorexmid | none | A form of excluded middle
df-ex | ~ exalim
alex | ~ alexim
exnal | ~ exnalim
alexn | ~ alexnim
exanali | ~ exanaliim
19.35 , 19.35ri | ~ 19.35-1
19.30 | none
19.39 | ~ i19.39
19.24 | ~ i19.24
19.36 , 19.36v | ~ 19.36-1
19.37 , 19.37v | ~ 19.37-1
19.32 | ~ 19.32r
19.31 | ~ 19.31r
exdistrf | ~ exdistrfor
exmo | ~ exmonim
mo2 | ~ mo2r , ~ mo3
nne | ~ nner , ~ nnedc
exmidne | ~ dcne
neor | ~ pm2.53 , ~ ori , ~ ord
neorian | ~ pm3.14
nnel | none | The reverse direction could be proved; the forward direction is double negation elimination.
nfrald | ~ nfraldxy , ~ nfraldya
rexnal | ~ rexnalim
rexanali | none
nrexralim | none
dfral2 | ~ ralexim
dfrex2 | ~ rexalim
nfrexd | ~ nfrexdxy , ~ nfrexdya
nfral | ~ nfralxy , ~ nfralya
nfra2 | ~ nfra1 , ~ nfralya
nfrex | ~ nfrexxy , ~ nfrexya
r19.30 | none
r19.35 | ~ r19.35-1
ralcom2 | ~ ralcom
2reuswap | ~ 2reuswapdc
df-pss and all proper subclass theorems | none | In set.mm, "A is a proper subclass of B" is defined to be ` ( A C_ B /\ A =/= B ) ` and this definition is almost always used in conjunction with excluded middle. A more natural definition might be ` ( A C_ B /\ E. x x e. ( B \ A ) )` , if we need proper subclass at all.
nss | ~ nssr
ddif | ~ ddifnel , ~ ddifss
dfss4 | ~ ssddif
n0f | ~ n0rf
n0 | ~ n0r , ~ fin0 , ~ fin0or
neq0 | ~ neq0r
reximdva0 | ~ reximdva0m
rabn0 | ~ rabn0m , ~ rabn0r
ssdif0 | ~ ssdif0im
inssdif0 | ~ inssdif0im
undif1 | ~ undif1ss
undif2 | ~ undif2ss
inundif | ~ inundifss
undif | ~ undifss
ssundif | ~ ssundifim
uneqdifeq | ~ uneqdifeqim
r19.2z | ~ r19.2m
r19.9rzv | ~ r19.9rmv
r19.45zv | ~ r19.45mv
dfif2 | ~ df-if
ifsb | ~ ifsbdc
dfif4 | none | Unused in set.mm
dfif5 | none | Unused in set.mm
ifeq1da | ~ ifeq1dadc
ifnot | none
ifan | none
ifor | none
ifeq2da | none
ifclda | ~ ifcldadc
ifeqda | none
elimif , ifbothda , ifid , eqif , ifval , elif , ifel , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD | none
ifboth | ~ ifbothdc
ifcl , ifcld | ~ ifcldcd , ~ ifcldadc
dedth , dedth2h , dedth3h , dedth4h , dedth2v , dedth3v , dedth4v , elimhyp , elimhyp2v , elimhyp3v , elimhyp4v , elimel , elimdhyp , keephyp , keephyp2v , keephyp3v , keepel | none | Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
ifex , ifexg | ~ ifcldcd , ~ ifcldadc
ifpr | none | Should be provable if the condition is decidable.
difsnid | ~ difsnss | One direction, for any set
~ nndifsnid | for a natural number
~ fidifsnid | for a finite set
iundif2 | ~ iundif2ss
iindif2 | ~ iindif2m
iinin2 | ~ iinin2m
iinin1 | ~ iinin1m
iinvdif | ~ iindif2m | Unused in set.mm
riinn0 | ~ riinm
riinrab | ~ iinrabm
iinuni | ~ iinuniss
iununi | ~ iununir
rintn0 | ~ rintm
trintss | ~ trintssm
ax-rep | ~ ax-coll | There are a lot of ways to state replacement and most/all of them hold, for example ~ zfrep6 or ~ funimaexg .
csbexg , csbex | ~ csbexga , ~ csbexa | set.mm uses case elimination to remove the ` A e. _V ` condition.
intex | ~ inteximm , ~ intexr | inteximm is the forward direction (but for inhabited rather than non-empty classes) and intexr is the reverse direction.
intexab | ~ intexabim
intexrab | ~ intexrabim
iinexg | ~ iinexgm | Changes not empty to inhabited
intabs | none | Lightly used in set.mm, and the set.mm proof is not intuitionistic
reuxfr2d , reuxfr2 , reuxfrd , reuxfr | none | The set.mm proof of reuxfr2d relies on 2reuswap
moabex | ~ euabex | In general, most of the set.mm ` E! ` theorems still hold, but a decent number of the ` E* ` ones get caught up on "there are two cases: the set exists or it does not"
snex | ~ snexg , ~ snex | The iset.mm version of ~ snex has an additional hypothesis
nssss | ~ nssssr
rmorabex | ~ euabex | See discussion under moabex
nnullss | ~ mss
opex | ~ opexg , ~ opex | The iset.mm version of ~ opex has additional hypotheses
otex | ~ otexg
df-so | ~ df-iso | Although we define ` Or ` to describe a weakly linear order (such as real numbers), there are some orders which are also trichotomous, for example ~ nntri3or , ~ pitri3or , and ~ nqtri3or .
sotric | ~ sotricim | One direction, for any weak linear order.
~ sotritric | For a trichotomous order.
~ nntri2 | For the specific order ` _E Or _om `
~ pitric | For the specific order `
~ nqtric | For the specific order `
sotrieq | ~ sotritrieq | For a trichotomous order
sotrieq2 | see sotrieq and then apply ~ ioran
issoi | ~ issod , ~ ispod | Many of the set.mm usages of issoi don't carry over, so there is less need for this convenience theorem.
isso2i | ~ issod | Presumably this could be proved if we need it.
df-fr | ~ df-frind
fri , dffr2 , frc | none | That any subset of the base set has an element which is minimal accordng to a well-founded relation presumably implies excluded middle (or is otherwise unprovable).
frss | ~ freq2 | Because the definition of ` Fr ` is different than set.mm, the proof would need to be different.
frirr | ~ frirrg | We do not yet have a lot of theorems for the case where ` A ` is a proper class.
fr2nr | none | Shouldn't be hard to prove if we need it (using a proof similar to ~frirrg and ~ en2lp ).
frminex | none | Presumably unprovable.
efrn2lp | none | Should be easy but lightly used in set.mm
dfepfr , epfrc | none | Presumably unprovable.
df-we | ~ df-wetr
wess | none | See frss entry. Holds for ` _E ` (see for example ~ wessep ).
weso | ~ wepo
wecmpep | none | ` We ` does not imply trichotomy in iset.mm
wefrc , wereu , wereu2 | none | Presumably not provable
dmxpid | ~ dmxpm
relimasn | ~ imasng
opswap | ~ opswapg
cnvso | ~ cnvsom
tz7.7 | none
ordelssne | none
ordelpss | none
ordsseleq , onsseleq | ~ onelss , ~ eqimss , ~ nnsseleq | Taken together, ~ onelss and ~ eqimss represent the reverse direction of the biconditional from ordsseleq . For natural numbers the biconditional is provable.
ordtri3or | ~ nntri3or | Ordinal trichotomy implies the law of the excluded middle as shown in ~ ordtriexmid .
ordtri2 | ~ nntri2 | ordtri2 for all ordinals presumably implies excluded middle although we don't have a specific proof analogous to ~ ordtriexmid .
ordtri3 , ordtri4 , ordtri2or3 , dford2 | none | Ordinal trichotomy implies the law of the excluded middle as shown in ~ ordtriexmid . We don't have similar proofs for every variation of of trichotomy but most of them are presumably powerful enough to imply excluded middle.
ordtri1 , ontri1 , onssneli , onssnel2i | ~ ssnel , ~ nntri1 | ~ ssnel is a trichotomy-like theorem which does hold, although it is an implication whereas ordtri1 is a biconditional. ~ nntri1 is biconditional, but just for natural numbers.
ordtr2 , ontr2 | none | Implies excluded middle as shown at ~ ontr2exmid
ordtr3 | none | This is weak linearity of ordinals, which presumably implies excluded middle by ~ ordsoexmid .
ord0eln0 , on0eln0 | ~ ne0i , ~ nn0eln0
nsuceq0 | ~ nsuceq0g
ordsssuc | ~ trsucss
ordequn | none | If you know which ordinal is larger, you can achieve a similar result via theorems such as ~ oneluni or ~ ssequn1 .
ordun | ~ onun2
ordtri2or | none | Implies excluded middle as shown at ~ ordtri2orexmid .
ordtri2or2 | ~ nntri2or2 | ordtri2or2 implies excluded middle as shown at ~ ordtri2or2exmid .
onsseli | none | See entry for ordsseleq
unizlim | none | The reverse direction is basically ~ uni0 plus ~ limuni
on0eqel | ~ 0elnn | The full on0eqel is conjectured to imply excluded middle by an argument similar to ~ ordtriexmid
snsn0non | none | Presumably would be provable (by first proving ` -. (/) e. { { (/) } } ` as in the set.mm proof, and then using that to show that ` { { (/) } } ` is not a transitive set).
onxpdisj | none | Unused in set.mm
onnev | none | Presumably provable (see snsn0non entry)
iotaex | ~iotacl , ~ euiotaex
dffun3 | ~ dffun5r
dffun5 | ~ dffun5r
dffv3 | ~ dffv3g
fvex | ~ funfvex | when evaluating a function within its domain
~ fvexg , ~ fvex | when the function is a set and is evaluated at a set
~ relrnfvex | when evaluating a relation whose range is a set
~ mptfvex | when the function is defined via maps-to, yields a set for all inputs, and is evaluated at a set
~ 1stexg , ~ 2ndexg | for the functions ` 1st ` and ` 2nd `
fvif | ~ fvifdc
ndmfv | ~ ndmfvg | The ` -. A e. _V ` case is ~ fvprc .
elfvdm | ~ relelfvdm
elfvex | ~ relelfvdm
fvmptss | ~ fvmptssdm
fvmptex | none | The set.mm proof relies on case elimination
fvmptnf | none | The set.mm proof relies on case elimination
fvmptn | none | The set.mm proof relies on case elimination
fvmptss2 | ~ fvmpt | What fvmptss2 adds is the cases where this is a proper class, or we are out of the domain.
fvopab4ndm | none
fndmdifeq0 | none | Although it seems like this might be intuitionizable, it is lightly used in set.mm.
f0cli | ~ ffvelrn
dff3 | ~ dff3im
dff4 | ~ dff4im
fvunsn | ~ fvunsng
funiunfv | ~ fniunfv , ~ funiunfvdm
funiunfvf | ~ funiunfvdmf
eluniima | ~ eluniimadm
dff14a , dff14b | none | The set.mm proof depends, in an apparently essential way, on excluded middle.
riotaex | ~ riotacl , ~ riotaexg
nfriotad | ~ nfriotadxy
csbriota , csbriotagOLD | ~ csbriotag
riotaxfrd | none | Although it may be intuitionizable, it is lightly used in set.mm. The set.mm proof relies on reuxfrd .
ovex | ~ fnovex | when the operation is a function evaluated within its domain.
~ ovexg | when the operation is a set and is evaluated at a set
~ relrnfvex | when the operation is a relation whose range is a set
~ mpt2fvex | When the operation is defined via maps-to, yields a set on any inputs, and is being evaluated at two sets.
fnov | ~ fnovim
ov3 | ~ ovi3 | Although set.mm's ov3 could be proved, it is only used a few places in set.mm, and in iset.mm those places need the modified form found in ~ ovi3 .
oprssdm | none
ndmovg , ndmov | ~ ndmfvg | These theorems are generally used in set.mm for case elimination which is why we just have the general ~ ndmfvg rather than an operation-specific version.
ndmovcl , ndmovcom , ndmovass , ndmovdistr , ndmovord , and ndmovordi | none | These theorems are generally used in set.mm for case elimination and the most straightforward way to avoid them is to add conditions that we are evaluating functions within their domains.
ndmovrcl | ~ elmpt2cl , ~ relelfvdm
caov4 | ~ caov4d | Note that ~ caov4d has a closure hypothesis.
caov411 | ~ caov411d | Note that ~ caov411d has a closure hypothesis.
caov42 | ~ caov42d | Note that ~ caov42d has a closure hypothesis.
caovdir | ~ caovdird | ~ caovdird adds some constraints about where the operations are evaluated.
caovdilem | ~ caovdilemd
caovlem2 | ~ caovlem2d
caovmo | ~ caovimo
ofval | ~ fnofval
offn , offveq , caofid0l , caofid0r , caofid1 , caofid2 | none | Assuming we really need to add conditions that the operations are functions being evaluated within their domains, there would be a fair bit of intuitionizing.
ordeleqon | none
ssonprc | none | not provable (we conjecture), but interesting enough to intuitionize anyway. ` U. A = On -> A e/ V ` is provable, and ` ( B e. On /\ U. A C_ B ) -> A e. V ` is provable. (One thing we presumably could prove is ` ( U. A C_ On /\ E. x x e. ( On \ U. A ) ) -> A e. V ` which might be easier to understand if we define (or think of) proper subset as meaning that the set difference is inhabited.)
onint | ~ onintss | onint implies excluded middle as shown in ~ onintexmid .
onint0 | none | Thought to be "trivially not intuitionistic", and it is not clear if there is an alternate way to state it that is true. If the empty set is in A then of course |^| A = (/), but the converse seems difficult. I don't know so much about the structure of the ordinals without linearity,
onssmin, onminesb, onminsb | none | Conjectured to not be provable without excluded middle, for the same reason as onint.
oninton | ~ onintonm
onintrab | none | The set.mm proof relies on the converse of ~ inteximm .
onintrab2 | ~ onintrab2im | The converse would appear to need the converse of ~ inteximm .
oneqmin | none | Falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.
onminex | none
onmindif2 | none | Conjectured to not be provable without excluded middle.
onmindif2 | none | Conjectured to not be provable without excluded middle.
ordpwsuc | ~ ordpwsucss | See the ~ ordpwsucss comment for discussion of the succcessor-like properites of ` ( ~P A i^i On ) ` . Full ordpwsuc implies excluded middle as seen at ~ ordpwsucexmid .
ordsucelsuc | ~ onsucelsucr , ~ nnsucelsuc | The converse of ~ onsucelsucr implies excluded middle, as shown at ~ onsucelsucexmid .
ordsucsssuc | ~ onsucsssucr , ~ nnsucsssuc | The converse of ~ onsucsssucr implies excluded middle, as shown at ~ onsucsssucexmid .
ordsucuniel | ~ sucunielr | Full ordsucuniel implies excluded middle, as shown at ~ ordsucunielexmid .
ordsucun | none yet | Conjectured to be provable in the reverse direction, but not the forward direction (implies some order totality).
ordunpr | none | Presumably not provable without excluded middle.
ordunel | none | Conjectured to not be provable (ordunel implies ordsucun).
onsucuni, ordsucuni | none | Conjectured to not be provable without excluded middle.
orduniorsuc | none | Presumably not provable.
ordunisuc | ~ onunisuci , ~ unisuc , ~ unisucg
orduniss2 | ~ onuniss2
0elsuc | none | This theorem may appear to be innocuous but it implies excluded middle as shown at ~ 0elsucexmid .
onuniorsuci | none | Conjectured to not be provable without excluded middle.
onuninsuci, orduninsuc | none | Conjectured to be provable in the forward direction but not the reverse one.
ordunisuc2 | ~ ordunisuc2r | The forward direction is conjectured to imply excluded middle. Here is a sketch of the proposed proof. Let om' be the set of all finite iterations of suc' A = ` ( ~P A i^i On ) ` on ` (/) `. (We can formalize this proof but not until we have om and at least finite induction.) Then om' = U. om' because if x e. om' then x = suc'^n (/) for some n, and then x C_ suc'^n (/) implies x e. suc'^(n+1) (/) e. om' so x e. U. om'. Now supposing the theorem, we know that A. x e. om' suc x e. om', so in particular 2o e. om', that is, 2o = suc'^n (/) for some n. (Note that 1o = suc' (/) - see ~ pw0 .) For n = 0 and n = 1 this is clearly false, and for n = m+3 we have 1o e. suc' suc' (/) , so 2o C_ suc' suc' (/), so 2o e. suc' suc' suc' (/) C_ suc' suc' suc' suc'^m (/) = 2o, contradicting ordirr. Thus 2o = suc' suc' (/) = suc' 1o. Applying this to X = ` { x e. { (/) } | ph } ` we have X C_ 1o implies X e. suc' 1o = 2o and hence X = (/) \/ X = 1o, and LEM follows (by ~ ordtriexmidlem2 for ` X = (/) ` and ~ rabsnt as seen in the ~ onsucsssucexmid proof for ` X = 1o ` ).
ordzsl, onzsl, dflim3, nlimon | none
dflim4 | ~ df-ilim | We conjecture that dflim4 is not equivalent to ~ df-ilim .
limsuc | none yet | Conjectured to be provable.
limsssuc | none yet | Conjectured to be provable.
tfinds | ~ tfis3
findsg | ~ uzind4 | findsg presumably could be proved, but there hasn't been a need for it.
xpexr2 | ~ xpexr2m
1stval | ~ 1stvalg
2ndval | ~ 2ndvalg
1stnpr | none | May be intuitionizable, but very lightly used in set.mm.
2ndnpr | none | May be intuitionizable, but very lightly used in set.mm.
brtpos | ~ brtposg
ottpos | ~ ottposg
ovtpos | ~ ovtposg
pwuninel | ~ pwuninel2 | The set.mm proof of pwuninel uses case elimination.
iunonOLD | ~ iunon
smofvon2 | ~ smofvon2dm
tfr1 | ~ tfri1
tfr2 | ~ tfri2
tfr3 | ~ tfri3
tfr2b , recsfnon , recsval | none | These transfinite recursion theorems are lightly used in set.mm.
df-rdg | ~ df-irdg | This definition combines the successor and limit cases (rather than specifying them as separate cases in a way which relies on excluded middle). In the words of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic", "we can still define many of the familiar set-theoretic operations by transfinite recursion on ordinals (see Aczel and Rathjen 2001, Section 4.2). This is fine as long as the definitions by transfinite recursion do not make case distinctions such as in the classical ordinal cases of successor and limit."
rdgfnon | ~ rdgifnon
ordge1n0 | ~ ordge1n0im , ~ ordgt0ge1
ondif1 | ~ dif1o | In set.mm, ondif1 is used for Cantor Normal Form
ondif2 , dif20el | none | The set.mm proof is not intuitionistic
brwitnlem | none | The set.mm proof is not intuitionistic
om0r | ~ om0 , ~ nnm0r
om00 | ~ nnm00
om00el | none
suc11reg | ~ suc11g
frfnom | ~ frecfnom | frecfnom adopts the ` frec ` notation and adds conditions on the characteristic function and initial value.
fr0g | ~ frec0g | frec0g adopts the ` frec ` notation and adds a condition on the characteristic function.
frsuc | ~ frecsuc | frecsuc adopts the ` frec ` notation and adds conditions on the characteristic function and initial value.
om0x | ~ om0
oaord1 | none yet
oaword | ~ oawordi | The other direction presumably could be proven but isn't yet.
omwordi | ~ nnmword | The set.mm proof of omwordi relies on case elimination.
omword1 | ~ nnmword
nnawordex | ~ nnaordex | nnawordex is only used a few places in set.mm
swoso | none | Unused in set.mm.
ecdmn0 | ~ ecdmn0m
erdisj, qsdisj, qsdisj2, uniinqs | none | These could presumably be restated to be provable, but they are lightly used in set.mm
xpider | ~ xpiderm
iiner | ~ iinerm
riiner | ~ riinerm
brecop2 | none | This is a form of reverse closure and uses excluded middle in its proof.
erov , erov2 | none | Unused in set.mm.
eceqoveq | none | Unused in set.mm.
df-sdom , relsdom , brsdom , dfdom2 , sdomdom , sdomnen , brdom2 , bren2 , domdifsn | none | Many aspects of strict dominance as developed in set.mm rely on excluded middle and a different definition would be needed if we wanted strict dominance to have the expected properties.
en1b | ~ en1bg
mapsnen , map1 , pw2f1olem , pw2f1o , pw2eng , pw2en | none | We have not added set exponentiation to iset.mm yet.
snfi | ~ snfig
difsnen | ~ fidifsnen
undom | none | The set.mm proof uses undif2 and we just have ~ undif2ss
xpdom3 | ~ xpdom3m
domunsncan | none | The set.mm proof relies on difsnen
omxpenlem , omxpen , omf1o | none | The set.mm proof relies on omwordi
enfixsn | none | The set.mm proof relies on difsnen
sbth and its lemmas, sbthb , sbthcl | ~ fisbth | The Schroeder-Bernstein Theorem implies excluded middle
2pwuninel | ~ 2pwuninelg
php | ~ phpm
snnen2o | ~ snnen2og , ~ snnen2oprc
onomeneq , onfin , onfin2 | none | The set.mm proofs rely on excluded middle
nnsdomo , sucdom2 , sucdom , 0sdom1dom , sdom1 | none | iset.mm doesn't yet have strict dominance
1sdom2 | ~ 1nen2 | Although the presence of ~ 1nen2 might make it look like a natural definition for strict dominance would be ` A ~<_ B /\ -. A ~~ B ` , that definition may be more suitable for finite sets than all sets, so at least for now we only define ` ~~ ` and express certain theorems (such as this one) in terms of equinumerosity which in set.mm are expressed in terms of strict dominance.
modom , modom2 | none | The set.mm proofs rely on excluded middle
1sdom , unxpdom , unxpdom2 , sucxpdom | none | iset.mm doesn't yet have strict dominance
pssinf | none | The set.mm proof relies on sdomnen
fisseneq | none | Perhaps it would work to generalize ~ en2eqpr from two element sets to any finite sets (presumably by induction) and thus prove fisseneq or something which easily implies it.
isinf | none | The set.mm proof uses the converse of ~ ssdif0im
fineqv | none | The set.mm proof relies on theorems we don't have, and even for the theorems we do have, we'd need to carefully look at what axioms they rely on.
pssnn | none | The set.mm proof uses excluded middle.
ssnnfi | none | The proof in ~ ssfiexmid would apply to this as well as to ssfi , since ` { (/) } e. _om `
ssfi | none | Implies excluded middle as shown at ~ ssfiexmid
domfi | none | Implies excluded middle as shown at ~ domfiexmid
xpfir | none | Nonempty would need to be changed to inhabited, but the set.mm proof also uses domfi
infi | none | Implies excluded middle as shown at ~ infiexmid
rabfi | none | Presumably the proof of ~ ssfiexmid could be adapted to show this implies excluded middle
finresfin | none | The set.mm proof is in terms of ssfi
f1finf1o | none | The set.mm proof is not intuitionistic
nfielex | none | The set.mm proof relies on neq0
en1eqsn , en1eqsnbi | none | The set.mm proof relies on fisseneq
diffi | ~ diffisn , ~ diffifi | diffi is not provable, as shown at ~ diffitest
enp1ilem , enp1i , en2 , en3 , en4 | none | The set.mm proof relies on excluded middle and undif1
findcard3 | none | The set.mm proof is in terms of strict dominance.
frfi | none | Not known whether this can be proved (either with the current ~ df-frind or any other possible concept analogous to ` Fr ` ).
fimax2g , fimaxg | none | The set.mm proof relies on frfi and fri
fisupg | none | The set.mm proof relies on excluded middle and presumably this theorem would need to be modified to be provable.
unfi | ~ unsnfi | For the union of a set and a singleton whose element is not a member of that set
~ unfidisj | For the union of two disjoint sets
for any two finite sets | Not provable according to Remark 8.1.17 of [AczelRathjen].
xpfi | none | This is Lemma 8.1.16 of [AczelRathjen] and is therefore presumably provable.
prfi | none | The set.mm proof depends on unfi and it would appear that mapping ` { A , B } ` to a natural number would decide whether ` A ` and ` B ` are equal and thus imply excluded middle.
fodomfi | none | Might be provable, for example via ~ ac6sfi or induction directly. The set.mm proof does use undom in addition to induction.
dmfi | ~ fundmfi
resfnfinfin | none | Presumably would need to have a ` B C_ A ` condition added, but lightly used in set.mm
residfi | none | Presumably provable, but lightly used in set.mm
cnvfi | ~ relcnvfi
rnfi | ~ funrnfi
fofi | ~ f1ofi | Presumably precluded by an argument similar to ~ domfiexmid (the set.mm proof relies on domfi).
abrexfi | none | At first glance it would appear that the mapping would need to be one to one or some other condition.
dfsup2 | none | The set.mm proof uses excluded middle in several places and the theorem is lightly used in set.mm.
supmo | ~ supmoti | The conditions on the order are different.
supexd , supex | none | The set.mm proof uses rmorabex
supeu | ~ supeuti
supval2 | ~ supval2ti
eqsup | ~ eqsupti
eqsupd | ~ eqsuptid
supcl | ~ supclti
supub | ~ supubti
suplub | ~ suplubti
suplub2 | ~ suplub2ti
supnub | none | Presumably provable, although the set.mm proof relies on excluded middle and it is not used until later in set.mm.
sup0riota , sup0 , infempty | none | Suitably modified verions may be provable, but they are unused in set.mm.
supmax | ~ supmaxti
fisup2g , fisupcl | none | Other variations may be possible, but the set.mm proof will not work as-is or with small modifications.
supgtoreq | none | The set.mm proof uses fisup2g and also trichotomy.
suppr | none | The formulation using ` if ` would seem to require a trichotomous order. For real numbers, supremum on a pair does yield the maximum of two numbers: see ~ maxcl , ~ maxle1 , ~ maxle2 , ~ maxleast , and ~ maxleb .
supiso | ~ supisoti
infexd , infex | none | See supexd
eqinf , eqinfd | ~ eqinfti , ~ eqinftid
infval | ~ infvalti
infcllem | ~ cnvinfex | infcllem has an unnecessary hypothesis; other than that these are the same
infcl | infclti
inflb | ~ inflbti
infglb | ~ infglbti
infglbb | none | Presumably provable with additional conditions (see suplub2)
infnlb | ~ infnlbti
infmin | ~ infminti
infmo | ~ infmoti
infeu | ~ infeuti
fimin2g , fiming | none | The set.mm proof relies on frfi and fri
fiinfg , fiinf2g | none | The set.mm proof relies on fiming
fiinfcl | none | See fisupcl
infltoreq | none | The set.mm proof depends on supgtoreq and fiinfcl
infpr | none | See suppr
infsn | ~ infsnti
infiso | ~ infisoti
ax-reg , axreg2 , zfregcl | ~ ax-setind | ax-reg implies excluded middle as seen at ~ regexmid
df-rank and all theorems related to the rank function | none | One possible definition is Definition 9.3.4 of [AczelRathjen], p. 91
df-aleph and all theorems involving aleph | none
df-cf and all theorems involving cofinality | none
df-acn and all theorems using this definition | none
cardf2 , cardon , isnum2 | ~ cardcl , ~ isnumi | It would also be easy to prove ` Fun card ` if there is a need.
ennum | none | The set.mm proof relies on isnum2
tskwe | none | Relies on df-sdom
xpnum | none | The set.mm proof relies on isnum2
cardval3 | ~ cardval3ex
cardid2 | none | The set.mm proof relies on onint
isnum3 | none
oncardid | none | The set.mm proof relies on cardid2
cardidm | none | Presumably this would need a condition on ` A ` but even with that, the set.mm proof relies on cardid2
oncard | none | The set.mm proof relies on theorems we don't have, and this theorem is unused in set.mm.
ficardom | none | Both the set.mm proof, and perhaps some possible alternative proofs, rely on onomeneq and perhaps other theorems we don't have currently.
ficardid | none | The set.mm proof relies on cardid2
cardnn | none | The set.mm proof relies on a variety of theorems we don't have currently.
cardnueq0 | none | The set.mm proof relies on cardid2
cardne | none | The set.mm proof relies on ordinal trichotomy (and if that can be solved there might be some more minor problems which require revisions to the theorem)
carden2a | none | The set.mm proof relies on excluded middle.
carden2b | ~ carden2bex
card1 , cardsn | none | Rely on a variety of theorems we don't currently have. Lightly used in set.mm.
carddomi2 | none | The set.mm proof relies on excluded middle.
sdomsdomcardi | none | Relies on a variety of theorems we don't currently have.
fodom , fodomnum | none | Presumably not provable as stated
entri3 | ~ fientri3 | Because full entri3 is equivalent to the axiom of choice, it implies excluded middle.
infinf | ~ infnfi | Defining "A is infinite" as ` _om ~<_ A ` follows definition 8.1.4 of [AczelRathjen], p. 71. It can presumably not be shown to be equivalent to ` -. A e. Fin ` in the absence of excluded middle.
df-wina , df-ina , df-tsk , df-gru , ax-groth and all theorems related to inaccessibles and large cardinals | none | For an introduction to inaccessibles and large set properties see Chapter 18 of [AczelRathjen], p. 165 (including why "large set properties" is more apt terminology than "large cardinal properties" in the absence of excluded middle).
addcompi | ~ addcompig
addasspi | ~ addasspig
mulcompi | ~ mulcompig
mulasspi | ~ mulasspig
distrpi | ~ distrpig
addcanpi | ~ addcanpig
mulcanpi | ~ mulcanpig
addnidpi | ~ addnidpig
ltapi | ~ ltapig
ltmpi | ~ ltmpig
nlt1pi | ~ nlt1pig
df-nq | ~ df-nqqs
df-nq | ~ df-nqqs
df-erq | none | Not needed given ~ df-nqqs
df-plq | ~ df-plqqs
df-mq | ~ df-mqqs
df-1nq | ~ df-1nqqs
df-ltnq | ~ df-ltnqqs
elpqn | none | Not needed given ~ df-nqqs
ordpipq | ~ ordpipqqs
addnqf | ~ dmaddpq , ~ addclnq | It should be possible to prove that ` +Q ` is a function, but so far there hasn't been a need to do so.
addcomnq | ~ addcomnqg
mulcomnq | ~ mulcomnqg
mulassnq | ~ mulassnqg
recmulnq | ~ recmulnqg
ltanq | ~ ltanqg
ltmnq | ~ ltmnqg
ltexnq | ~ ltexnqq
archnq | ~ archnqq
df-np | ~ df-inp
df-1p | ~ df-i1p
df-plp | ~ df-iplp
df-ltp | ~ df-iltp
elnp , elnpi | ~ elinp
prn0 | ~ prml , ~ prmu
prpssnq | ~ prssnql , ~ prssnqu
elprnq | ~ elprnql , ~ elprnqu
prcdnq | ~ prcdnql , ~ prcunqu
prub | ~ prubl
prnmax | ~ prnmaxl
npomex | none
prnmadd | ~ prnmaddl
genpv | ~ genipv
genpcd | ~ genpcdl
genpnmax | ~ genprndl
ltrnq | ~ltrnqg , ~ ltrnqi
genpcl | ~ addclpr , ~ mulclpr
genpass | ~ genpassg
addclprlem1 | ~ addnqprllem , ~ addnqprulem
addclprlem2 | ~ addnqprl , ~ addnqpru
plpv | ~ plpvlu
mpv | ~ mpvlu
nqpr | ~ nqprlu
mulclprlem | ~ mulnqprl , ~ mulnqpru
addcompr | ~ addcomprg
addasspr | ~ addassprg
mulcompr | ~ mulcomprg
mulasspr | ~ mulassprg
distrlem1pr | ~ distrlem1prl , ~ distrlem1pru
distrlem4pr | ~ distrlem4prl , ~ distrlem4pru
distrlem5pr | ~ distrlem5prl , ~ distrlem5pru
distrpr | ~ distrprg
ltprord | ~ ltprordil | There hasn't yet been a need to investigate versions which are biconditional or which involve proper subsets.
psslinpr | ~ ltsopr
prlem934 | ~ prarloc2
ltaddpr2 | ~ ltaddpr
ltexprlem1 , ltexprlem2 , ltexprlem3 , ltexprlem4 | none | See the lemmas for ~ ltexpri generally.
ltexprlem5 | ~ ltexprlempr
ltexprlem6 | ~ ltexprlemfl , ~ ltexprlemfu
ltexprlem7 | ~ ltexprlemrl , ~ ltexprlemru
ltapr | ~ ltaprg
addcanpr | ~ addcanprg
prlem936 | ~ prmuloc2
reclem2pr | ~ recexprlempr
reclem3pr | ~ recexprlem1ssl , ~ recexprlem1ssu
reclem4pr | ~ recexprlemss1l , ~ recexprlemss1u , ~ recexprlemex
supexpr , suplem1pr , suplem2pr | ~ caucvgprpr | The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
mulcmpblnrlem | ~ mulcmpblnrlemg
ltsrpr | ~ ltsrprg
dmaddsr , dmmulsr | none | Although these presumably could be proved in a way similar to ~ dmaddpq and ~dmmulpq (in fact ~ dmaddpqlem would seem to be easily generalizable to anything of the form ` ( ( S X. T ) /. R ) ` ), we haven't yet had a need to do so.
addcomsr | ~ addcomsrg
addasssr | ~ addasssrg
mulcomsr | ~ mulcomsrg
mulasssr | ~ mulasssrg
distrsr | ~ distrsrg
ltasr | ~ ltasrg
sqgt0sr | ~ mulgt0sr , ~ apsqgt0
recexsr | ~ recexsrlem | This would follow from sqgt0sr (as in the set.mm proof of recexsr), but "not equal to zero" would need to be changed to "apart from zero".
mappsrpr , ltpsrpr , map2psrpr | ~ prsrpos , ~ prsrlt , ~ srpospr
supsrlem , supsr | ~ caucvgsr | The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
axaddf , ax-addf , axmulf , ax-mulf | none | Because these are described as deprecated in set.mm, we haven't figured out what would be involved in proving them for iset.mm.
ax1ne0 , ax-1ne0 | ~ ax0lt1 , ~ ax-0lt1 , ~ 1ap0 , ~ 1ne0
axrrecex , ax-rrecex | ~ axprecex , ~ ax-precex
axpre-lttri , ax-pre-lttri | ~ axpre-ltirr , ~ axpre-ltwlin , ~ ax-pre-ltirr , ~ ax-pre-ltwlin
axpre-sup , ax-pre-sup , axsup | ~ axcaucvg , ~ ax-caucvg , ~ caucvgre | The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
elimne0 | none | Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
xrltnle | ~ xrlenlt
ssxr | ~ df-xr | Lightly used in set.mm
ltnle , ltnlei , ltnled | ~ lenlt , ~ zltnle
lttri2 , lttri2i , lttri2d | ~ qlttri2 | Real number trichotomy is not provable.
lttri4 | ~ ztri3or , ~ qtri3or | Real number trichotomy is not provable.
leloe , eqlelt , leloei , leloed , eqleltd | none
leltne , leltned | ~ leltap , ~ leltapd
ltneOLD | ~ ltne , ~ ltap
letric , letrii , letrid | ~ zletric , ~ qletric
ltlen , ltleni , ltlend | ~ ltleap , ~ zltlen , ~ qltlen
ne0gt0 , ne0gt0d | ~ ap0gt0 , ~ ap0gt0d
lecasei , ltlecasei | none | These are real number trichotomy
lelttric | ~ zlelttric , ~ qlelttric
lttrid , lttri4d | none | These are real number trichotomy
leneltd | ~ leltapd
dedekind , dedekindle | none
mul02lem1 | none | The one use in set.mm is not needed in iset.mm.
negex | ~ negcl
msqgt0 , msqgt0i , msqgt0d | ~ apsqgt0 | "Not equal to zero" is changed to "apart from zero"
relin01 | none | Relies on real number trichotomy.
ltordlem , ltord1 , leord1 , eqord1 , ltord2 , leord2 , eqord2 | none | Although these presumably could be proved using theorems like ~ letri3 and ~ lenlt , at least for now we have chosen to just invoke those other theorems directly (example: ~ expcan and its lemma ~ expcanlem ) which avoids some extra set variables and produces proofs which are almost as short.
wloglei , wlogle | none | These depend on real number trichotomy and are not used until later in set.mm.
recex | ~ recexap | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcand, mulcan2d | ~ mulcanapd , ~ mulcanap2d | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcanad , mulcan2ad | ~ mulcanapad , ~ mulcanap2ad | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcan , mulcan2 , mulcani | ~ mulcanap , ~ mulcanap2 , ~ mulcanapi | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mul0or , mul0ori , mul0ord | none | Remark 2.19 of [Geuvers] says that this does not hold in general and has a counterexample.
mulne0b , mulne0bd , mulne0bad , mulne0bbd | ~ mulap0b , ~ mulap0bd , ~ mulap0bad , ~ mulap0bbd
mulne0 , mulne0i , mulne0d | ~ mulap0 , ~ mulap0i , ~ mulap0d
receu | ~ receuap | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulnzcnopr | none
msq0i , msq0d | ~ sqeq0 , ~ sqeq0i | These slight restatements of sqeq0 are unused in set.mm.
mulcan1g , mulcan2g | various cancellation theorems | Presumably this is unavailable for the same reason that mul0or is unavailable.
1div0 | none | This could be proved, but the set.mm proof does not work as-is.
divval | ~ divvalap | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divmul , divmul2 , divmul3 | ~ divmulap , ~ divmulap2 , divmulap3 | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcl , reccl | ~ divclap , ~ recclap | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcan1 , divcan2 | ~ divcanap1 , ~ divcanap2 | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
diveq0 | ~ diveqap0 | In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divne0b , divne0 | ~ divap0b , ~ divap0
recne0 | ~ recap0
recid , recid2 | ~ recidap , ~ recidap2
divrec | ~ divrecap
divrec2 | ~ divrecap2
divass | ~ divassap
div23 , div32 , div13 , div12 | ~ div23ap , ~ div32ap , ~ div13ap , ~ div12ap
divmulass | ~ divmulassap
divmulasscom | ~ divmulasscomap
divdir , divcan3 , divcan4 | ~ divdirap , ~ divcanap3 , ~ divcanap4
div11 , divid , div0 | ~ div11ap , ~ dividap , ~ div0ap
diveq1 , divneg | ~ diveqap1 , ~ divnegap
muldivdir | ~ muldivdirap
divsubdir | ~ divsubdirap
recrec , rec11 , rec11r | ~ recrecap , ~ rec11ap , ~ rec11rap
divmuldiv , divdivdiv , divcan5 | ~ divmuldivap , ~ divdivdivap , ~ divcanap5
divmul13 , divmul24 , divmuleq | ~ divmul13ap , ~ divmul24ap , ~ divmuleqap
recdiv , divcan6 , divdiv32 , divcan7 | ~ recdivap , ~ divcanap6 , ~ divdiv32ap , ~ divcanap7
dmdcan , divdiv1 , divdiv2 , recdiv2 | ~ dmdcanap , ~ divdivap1 , ~ divdivap2 , ~ recdivap2
ddcan , divadddiv , divsubdiv | ~ ddcanap , ~ divadddivap , ~ divsubdivap
ddcan , divadddiv , divsubdiv | ~ ddcanap , ~ divadddivap , ~ divsubdivap
conjmul , rereccl, redivcl | ~ conjmulap , ~ rerecclap , ~ redivclap
div2neg , divneg2 | ~ div2negap , ~ divneg2ap
recclzi , recne0zi , recidzi | ~ recclapzi , ~ recap0apzi , ~ recidapzi
reccli , recidi , recreci | ~ recclapi , ~ recidapi , ~ recrecapi
dividi , div0i | ~ dividapi , ~ div0api
divclzi , divcan1zi , divcan2zi | ~ divclapzi , ~ divcanap1zi , ~ divcanap2zi
divreczi , divcan3zi , divcan4zi | ~ divrecapzi , ~ divcanap3zi , ~ divcanap4zi
rec11i , rec11ii | ~ rec11api , ~ rec11apii
divcli , divcan2i , divcan1i , divreci , divcan3i , divcan4i | ~ divclapi , ~ divcanap2i , ~ divcanap1i , ~ divrecapi , ~ divcanap3i , ~ divcanap4i
div0i | ~ divap0i
divasszi , divmulzi , divdirzi , divdiv23zi | ~ divassapzi , ~ divmulapzi , ~ divdirapzi , ~ divdiv23apzi
divmuli , divdiv32i | ~ divmulapi , ~ divdiv32api
divassi , divdiri , div23i , div11i | ~ divassapi , ~ divdirapi , ~ div23api , ~ div11api
divmuldivi, divmul13i, divadddivi, divdivdivi | ~ divmuldivapi , ~ divmul13api , ~ divadddivapi , ~ divdivdivapi
rerecclzi , rereccli , redivclzi , redivcli | ~ rerecclapzi , ~ rerecclapi , ~ redivclapzi , ~ redivclapi
reccld , rec0d , recidd , recid2d , recrecd , dividd , div0d | ~ recclapd , ~ recap0d , ~ recidapd , ~ recidap2d , ~ recrecapd , ~ dividapd , ~ div0apd
divcld , divcan1d , divcan2d , divrecd , divrec2d , divcan3d , divcan4d | ~ divclapd , ~ divcanap1d , ~ divcanap2d , ~ divrecapd , ~ divrecap2d , ~ divcanap3d , ~ divcanap4d
diveq0d , diveq1d , diveq1ad , diveq0ad , divne1d , div0bd , divnegd , divneg2d , div2negd | ~ diveqap0d , ~ diveqap1d , ~ diveqap1ad , ~ diveqap0ad , ~ divap1d , ~ divap0bd , ~ divnegapd , ~ divneg2apd , ~ div2negapd
divne0d , recdivd , recdiv2d , divcan6d , ddcand , rec11d | ~ divap0d , ~ recdivapd , ~ recdivap2d , ~ divcanap6d , ~ ddcanapd , ~ rec11apd
divmuld , div32d , div13d , divdiv32d , divcan5d , divcan5rd , divcan7d , dmdcand , dmdcan2d , divdiv1d , divdiv2d | ~ divmulapd , ~ div32apd , ~ div13apd , ~ divdiv32apd , ~ divcanap5d , ~ divcanap5rd , ~ divcanap7d , ~ dmdcanapd , ~ dmdcanap2d , ~ divdivap1d , ~ divdivap2d
divmul2d, divmul3d, divassd, div12d, div23d, divdird, divsubdird, div11d | ~ divmulap2d , ~ divmulap3d , ~ divassapd , ~ div12apd , ~ div23apd , ~ divdirapd , ~ divsubdirapd , ~ div11apd
divmuldivd | ~ divmuldivapd
rereccld , redivcld | ~ rerecclapd , ~ redivclapd
mvllmuld | ~ mvllmulapd
elimgt0 , elimge0 | none | Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
mulge0b , mulsuble0b | none | Presumably unprovable for reasons analogous to mul0or.
mulle0b | ~ mulle0r | The converse of mulle0r is presumably unprovable for reasons analogous to mul0or.
ledivp1i , ltdivp1i | none | Presumably could be proved, but unused in set.mm.
fimaxre | ~ fimaxre2 | When applied to a pair this could show which of two unequal real numbers is larger, so perhaps not provable for that reason. (see ~ fin0 for inhabited versus nonempty).
fimaxre3 | none | The set.mm proof relies on abrexfi .
fiminre | none | See fimaxre
sup2 , sup3 , sup3ii | none | We won't be able to have the least upper bound property for all nonempty bounded sets. In cases where we can show that the supremum exists, we might be able to prove slightly different ways of stating there is a supremum.
infm3 | none | See sup3
suprcl , suprcld , suprclii | ~ supclti
suprub , suprubd , suprubii | ~ suprubex
suprlub , suprlubii | ~ suprlubex
suprnub , suprnubii | ~ suprnubex
suprleub , suprleubii | ~ suprleubex
supaddc , supadd | none | Presumably provable with suitable conditions for the existence of the supremums
supmul1 , supmul | none | Presumably provable with suitable conditions for the existence of the supremums
riotaneg | none | The theorem is unused in set.mm and the set.mm proof relies on reuxfrd
infrecl | ~ infclti
infrenegsup | ~ infrenegsupex
infregelb | none yet | Presumably could be handled in a way analogous to ~ suprleubex
infrelb | none yet | Presumably could be handled in a way analogous to ~ suprubex
supfirege | ~ suprubex | The question here is whether results like ~ maxle1 can be generalized (presumably by induction) from pairs to finite sets.
crne0 | ~ crap0
ofsubeq0 , ofnegsub , ofsubge0 | none | Depend on ofval and/or offn .
df-nn | ~ dfnn2
dfnn3 | ~ dfnn2 | Presumably could be proved, as it is a slight variation of dfnn2
avgle | ~ qavgle
nnunb | none | Presumably provable from ~ arch but unused in set.mm.
frnnn0supp , frnnn0fsupp | ~ nn0supp | iset.mm does not yet have either the notation, or in some cases the theorems, related to the support of a function or a fintely supported function.
suprzcl | ~ suprzclex
zriotaneg | none | Lightly used in set.mm
suprfinzcl | none
decex | ~ deccl
halfthird | none | Presumably will be easy to intuitionize
5recm6rec | none | Presumably will be easy to intuitionize
uzwo , uzwo2 , nnwo , nnwof , nnwos | none | Presumably would imply excluded middle, unless there is something which makes this different from ~ nnregexmid .
negn0 | ~ negm
uzinfi, nninf, nn0inf | none | Presumably provable
infssuzle | ~ infssuzledc
infssuzcl | ~ infssuzcldc
supminf | ~ supminfex
zsupss , suprzcl2 | ~ zsupcl , ~ suprzclex
suprzub | none | Presumably could prove something like this with different conditions for the existence of the supremum (see ~ infssuzledc for something along these lines).
uzsupss | ~ zsupcl
uzwo3 , zmin | none | Proved in terms of supremum theorems and presumably not possible without excluded middle.
zbtwnre | none | Proved in terms of supremum theorems and presumably not possible without excluded middle.
rebtwnz | ~ qbtwnz
rpneg | ~ rpnegap
xrlttri , xrlttri2 | none | A generalization of real trichotomy.
xrleloe , xrleltne , dfle2 | none | Consequences of real trichotomy.
xrltlen | none | We presumably could prove an analogue to ~ ltleap but we have not yet defined apartness for extended reals (` # ` is for complex numbers).
dflt2 | none
xrletri | none
xrmax1 , xrmax2 , xrmin1 , xrmin2 , xrmaxeq , xrmineq , xrmaxlt , xrltmin , xrmaxle , xrlemin , max0sub , ifle | none
max1 | ~ maxle1
max2 | ~ maxle2
2resupmax | none | Proved from real trichotomy. We express maximum in iset.mm using ` sup ( { A , B } , RR , < ) ` rather than ` if ( A <_ B , B , A ) ` . The former has the expected maximum properties such as ~ maxcl , ~ maxle1 , ~ maxle2 , ~ maxleast , and ~ maxleb .
min1 | ~ min1inf
min2 | ~ min2inf
maxle | ~ maxleastb
lemin | ~ lemininf
maxlt | ~ maxltsup
ltmin | ~ ltmininf
qsqueeze | none yet | Presumably provable from ~ qbtwnre and ~ squeeze0 , but unused in set.mm.
qextltlem , qextlt , qextle | none | The set.mm proof is not intuitionistic.
xralrple , alrple | none yet | Now that we have ~ qbtwnxr , it looks like the set.mm proof would work with minor changes.
xnegex | ~ xnegcl
xaddval , xaddf , xmulval , xaddpnf1 , xaddpnf2 , xaddmnf1 , xaddmnf2 , pnfaddmnf , mnfaddpnf , rexadd , rexsub , xaddnemnf , xaddnepnf , xnegid , xaddcl , xaddcom , xaddid1 , xaddid2 , xnegdi , xaddass , xaddass2 , xpncan , xnpcan , xleadd1a , xleadd2a , xleadd1 , xltadd1 , xltadd2 , xaddge0 , xle2add , xlt2add , xsubge0 , xposdif , xlesubadd , xmullem , xmullem2 , xmulcom , xmul01 , xmul02 , xmulneg1 , xmulneg2 , rexmul , xmulf , xmulcl , xmulpnf1 , xmulpnf2 , xmulmnf1 , xmulmnf2 , xmulpnf1n , xmulid1 , xmulid2 , xmulm1 , xmulasslem2 , xmulgt0 , xmulge0 , xmulasslem , xmulasslem3 , xmulass , xlemul1a , xlemul2a , xlemul1 , xlemul2 , xltmul1 , xltmul2 , xadddilem , xadddi , xadddir , xadddi2 , xadddi2r , x2times , xaddcld , xmulcld , xadd4d | none | There appears to be no fundamental obstacle to proving these, because disjunctions can arise from ~ elxr rather than excluded middle. However, ` -e ` , ` +e ` , and ` *e ` are lightly used in set.mm, and the set.mm proofs would require significant changes.
ixxub , ixxlb | none
iccen | none
supicc , supiccub , supicclub , supicclub2 | none
ixxun , ixxin | none
ioon0 | ~ ioom | Non-empty is changed to inhabited
iooid | ~ iooidg
ndmioo | none | See discussion at ndmov but set.mm uses excluded middle, both in proving this and in using it.
lbioo , ubioo | ~ lbioog , ~ ubioog
iooin | none
icc0 | ~ icc0r
ioorebas | ~ ioorebasg
ge0xaddcl , ge0xmulcl | none | Rely on xaddcl and xmulcl ; see discussion in this list for those theorems.
icoun , snunioo , snunico , snunioc , prunioo | none
ioojoin | none
difreicc | none
iccsplit | none | This depends, apparently in an essential way, on real number trichotomy.
xov1plusxeqvd | none | This presumably could be proved if not equal is changed to apart, but is lightly used in set.mm.
fzn0 | ~ fzm
fz0 | none | Although it would be possible to prove a version of this with the additional conditions that ` M e. _V ` and ` N e. _V ` , the theorem is lightly used in set.mm.
fzon0 | ~ fzom
fzo0n0 | ~ fzo0m
ssfzoulel | none | Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
fzonfzoufzol | none | Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
elfznelfzo , elfznelfzob , injresinjlem , injresinj | none | Some or all of this presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
flcl , reflcl , flcld | ~ flqcl , ~ flqcld
fllelt | ~ flqlelt
flle | ~ flqle
flltp1 , fllep1 | ~ flqltp1
fraclt1 , fracle1 | ~ qfraclt1
fracge0 | ~ qfracge0
flge | ~ flqge
fllt | ~ flqlt
flflp1 | none | The set.mm proof relies on case elimination.
flidm | ~ flqidm
flidz | ~ flqidz
flltnz | ~ flqltnz
flwordi | ~ flqwordi
flword2 | ~ flqword2
flval2 , flval3 | none | Unused in set.mm
flbi | ~ flqbi
flbi2 | ~ flqbi2
ico01fl0 | none | Presumably could be proved for rationals, but lightly used in set.mm.
flge0nn0 | ~ flqge0nn0
flge1nn | ~ flqge1nn
refldivcl | ~ flqcl
fladdz | ~ flqaddz
flzadd | ~ flqzadd
flmulnn0 | ~ flqmulnn0
fldivle | ~ flqle
ltdifltdiv | none | Unused in set.mm.
fldiv4lem1div2uz2 , fldiv4lem1div2 | none | Presumably provable, but lightly used in set.mm.
ceilval | ~ ceilqval | The set.mm ceilval, with a real argument and no additional conditions, is probably provable if there is a need.
dfceil2 , ceilval2 | none | Unused in set.mm.
ceicl | ~ ceiqcl
ceilcl | ~ ceilqcl
ceilge | ~ ceilqge
ceige | ~ ceiqge
ceim1l | ~ ceiqm1l
ceilm1lt | ~ ceilqm1lt
ceile | ~ ceiqle
ceille | ~ ceilqle
ceilidz | ~ ceilqidz
flleceil | ~ flqleceil
fleqceilz | ~ flqeqceilz
quoremz , quoremnn0 , quoremnn0ALT | none | Unused in set.mm.
intfrac2 | ~ intqfrac2
fldiv | ~ flqdiv
fldiv2 | none | Presumably would be provable if real is changed to rational.
fznnfl | none | Presumably would be provable if real is changed to rational.
uzsup , ioopnfsup , icopnfsup , rpsup , resup , xrsup | none | As with most theorems involving supremums, these would likely need significant changes
modval | ~ modqval | As with theorems such as ~ flqcl , we prove most of the modulo related theorems for rationals, although other conditions on real arguments other than whether they are rational would be possible in the future.
modvalr | ~ modqvalr
modcl , modcld | ~ modqcl , ~ modqcld
flpmodeq | ~ flqpmodeq
mod0 | ~ modq0
mulmod0 | ~ mulqmod0
negmod0 | ~ negqmod0
modge0 | ~ modqge0
modlt | ~ modqlt
modelico | ~ modqelico
moddiffl | ~ modqdiffl
moddifz | ~ modqdifz
modfrac | ~ modqfrac
flmod | ~ flqmod
intfrac | ~ intqfrac
modmulnn | ~ modqmulnn
modvalp1 | ~ modqvalp1
modid | ~ modqid
modid0 | ~ modqid0
modid2 | ~ modqid2
0mod | ~ q0mod
1mod | ~ q1mod
modabs | ~ modqabs
modabs2 | ~ modqabs2
modcyc | ~ modqcyc
modcyc2 | ~ modqcyc2
modadd1 | ~ modqadd1
modaddabs | ~ modqaddabs
modaddmod | ~ modqaddmod
muladdmodid | ~ mulqaddmodid
modmuladd | ~ modqmuladd
modmuladdim | ~ modqmuladdim
modmuladdnn0 | ~ modqmuladdnn0
negmod | ~ qnegmod
modadd2mod | ~ modqadd2mod
modm1p1mod0 | ~ modqm1p1mod0
modltm1p1mod | ~ modqltm1p1mod
modmul1 | ~ modqmul1
modmul12d | ~ modqmul12d
modnegd | ~ modqnegd
modadd12d | ~ modqadd12d
modsub12d | ~ modqsub12d
modsubmod | ~ modqsubmod
modsubmodmod | ~ modqsubmodmod
2txmodxeq0 | ~ q2txmodxeq0
2submod | ~ q2submod
modmulmod | ~ modqmulmod
modmulmodr | ~ modqmulmodr
modaddmulmod | ~ modqaddmulmod
moddi | ~ modqdi
modsubdir | ~ modqsubdir
modeqmodmin | ~ modqeqmodmin
modirr | none | A version of this (presumably modified) may be possible, but it is unused in set.mm
om2uz0i | ~ frec2uz0d
om2uzsuci | ~ frec2uzsucd
om2uzuzi | ~ frec2uzuzd
om2uzlti | ~ frec2uzltd
om2uzlt2i | ~ frec2uzlt2d
om2uzrani | ~ frec2uzrand
om2uzf1oi | ~ frec2uzf1od
om2uzisoi | ~ frec2uzisod
om2uzoi , ltweuz , ltwenn , ltwefz | none | Based on theorems like ~ nnregexmid it is not clear what, if anything, along these lines is possible.
om2uzrdg | ~ frec2uzrdg
uzrdglem | ~ frecuzrdglem
uzrdgfni | ~ frecuzrdgfn
uzrdg0i | ~ frecuzrdg0
uzrdgsuci | ~ frecuzrdgsuc
uzinf | none | See ominf
uzrdgxfr | none | Presumably could be proved if restated in terms of ` frec ` (a la ~ frec2uz0d ). However, it is lightly used in set.mm.
fzennn | ~ frecfzennn
fzen2 | ~ frecfzen2
cardfz | none | Cardinality does not work the same way without excluded middle and iset.mm has few cardinality related theorems.
hashgf1o | ~ frechashgf1o
fzfi | ~ fzfig
fzfid | ~ fzfigd
fzofi | ~ fzofig
fsequb | none | Seems like it might be provable, but unused in set.mm
fsequb2 | none | The set.mm proof does not work as-is
fseqsupcl | none | The set.mm proof relies on fisupcl and it is not clear whether this supremum theorem or anything similar can be proved.
fseqsupubi | none | The set.mm proof relies on fsequb2 and suprub and it is not clear whether this supremum theorem or anything similar can be proved.
uzindi | none | This could presumably be proved, perhaps from ~ uzsinds , but is lightly used in set.mm
axdc4uz | none | Although some versions of constructive mathematics accept dependent choice, we have not yet developed it in iset.mm
ssnn0fi , rabssnn0fi | none | Conjectured to imply excluded middle along the lines of ~ nnregexmid or ~ ssfiexmid
df-seq | ~ df-iseq
seqex | ~ iseqex | The only difference is the differing syntax of ` seq ` .
seqeq1 , seqeq2 , seqeq3 , seqeq1d , seqeq2d , seqeq3d , seqeq123d | ~ iseqeq1 , ~ iseqeq2 , ~ iseqeq3
nfseq | ~ nfiseq
seqval | ~ iseqval
seqfn | ~ iseqfn
seq1 , seq1i | ~ iseq1
seqp1 , seqp1i | ~ iseqp1
seqm1 | ~ iseqm1
seqcl2 | none yet | Presumably could prove this (with adjustments analogous to ~ iseqp1 ). The third argument to ` seq ` in iset.mm would correspond to ` C ` rather than ` D ` (some of the other ` seq ` related theorems do not allow ` C ` and ` D ` to be different, but seqcl2 does).
seqf2 | none yet | Presumably could prove this, analogously to seqcl2.
seqcl | ~ iseqcl | iseqcl requires that ` F ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seqcl . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf | ~ iseqf
seqfveq2 | ~ iseqfveq2
seqfeq2 | ~ iseqfeq2
seqfveq | ~ iseqfveq
seqfeq | ~ iseqfeq
seqshft2 | ~ iseqshft2
seqres | none | Should be intuitionizable as with the other ` seq ` theorems, but unused in set.mm
serf | ~ iserf
serfre | ~ iserfre
sermono | ~ isermono | isermono requires that ` F ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in sermono . Given that the intention is to use this for infinite series, there may be no need to look into whether this requirement can be relaxed.
seqsplit | ~ iseqsplit | iseqsplit requires that ` F ` be defined on ` ( ZZ>= `` K ) ` not merely ` ( K ... N ) ` as in seqsplit . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seq1p | ~ iseq1p | iseq1p requires that ` F ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seq1p . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr3 | ~ iseqcaopr3 | iseqcaopr3 requires that ` F ` , ` G ` , and ` H ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seqcaopr3 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr2 | ~ iseqcaopr2 | iseqcaopr2 requires that ` F ` , ` G ` , and ` H ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seqcaopr2 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr | ~ iseqcaopr | iseqcaopr requires that ` F ` , ` G ` , and ` H ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seqcaopr . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf1o | none | May be adaptable with sufficient attention to issues such as whether the operation ` .+ ` is closed with respect to ` C ` as well as ` S ` , and perhaps other issues.
seradd | ~ iseradd | iseradd requires that ` F ` , ` G ` , and ` H ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in seradd . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
sersub | ~ isersub | isersub requires that ` F ` , ` G ` , and ` H ` be defined on ` ( ZZ>= `` M ) ` not merely ` ( M ... N ) ` as in sersub . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqid3 | ~ iseqid3 , ~ iseqid3s | ~ iseqid3 and ~ iseqid3s differ with respect to which entries in ` F ` need to be zero. Further refinements to the hypotheses including where ` F ` needs to be defined and the like might be possible (with a greater amount of work), but perhaps these are sufficient.
seqid | ~ iseqid
seqid2 | none yet | It should be possible to come up with a (presumably modified) version of this, but we have not done so yet.
seqhomo | ~ iseqhomo
seqz | ~ iseqz
seqfeq4 , seqfeq3 | none yet | It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
seqdistr | ~ iseqdistr
ser0 | ~ iser0 | The only difference is the syntax of ` seq ` .
ser0f | ~ iser0f | The only difference is the syntax of ` seq ` .
serge0 | ~ serige0 | This theorem uses ` CC ` as the final argument to ` seq ` which probably will be more convenient even if all the values are elements of a subset of ` CC ` .
serle | ~ serile | This theorem uses ` CC ` as the final argument to ` seq ` which probably will be more convenient even if all the values are elements of a subset of ` CC ` .
ser1const , seqof , seqof2 | none yet | It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
df-exp | ~ df-iexp | The difference is that ` seq ` in iset.mm takes three arguments compared with two in set.mm.
expval | ~ expival
expnnval | ~ expinnval
expneg | ~ expnegap0 | The set.mm theorem does not exclude the case of dividing by zero.
expneg2 | ~ expineg2
expn1 | ~ expn1ap0
expcl2lem | ~ expcl2lemap
reexpclz | ~ reexpclzap
expclzlem | ~ expclzaplem
expclz | ~ expclzap
expne0 | ~ expap0
expne0i | ~ expap0i
expnegz | ~ expnegzap
mulexpz | ~ mulexpzap
exprec | ~ exprecap
expaddzlem , expaddz | ~ expaddzaplem , ~ expaddzap
expmulz | ~ expmulzap
expsub | ~ expsubap
expp1z | ~ expp1zap
expm1 | ~ expm1ap
expdiv | ~ expdivap
ltexp2 , leexp2 , leexp2 , ltexp2d , leexp2d | none | Presumably provable, but the set.mm proof uses ltord1
ltexp2r , ltexp2rd | none | Presumably provable, but the set.mm proof uses ltexp2
sqdiv | ~ sqdivap
sqgt0 | ~ sqgt0ap
sqrecii , sqrecd | ~ exprecap
sqdivi | ~ sqdivapi
sqgt0i | ~ sqgt0api
sqlecan | ~ lemul1 | Unused in set.mm
sqeqori | none | The reverse direction is ~ oveq1 together with ~ sqneg . The forward direction is presumably not provable, see mul0or for more discussion.
subsq0i , sqeqor | none | Variations of sqeqori .
sq01 | none | Lightly used in set.mm. Presumably not provable as stated, for reasons analogous to mul0or .
crreczi | none | Presumably could be proved if not-equal is changed to apart, but unused in set.mm.
expmulnbnd | none | Should be possible to prove this or something similar, but the set.mm proof relies on case elimination based on whether ` 0 <_ A ` or not.
digit2 , digit1 | none | Depends on modulus and floor, and unused in set.mm.
modexp | none | Depends on modulus. Presumably it, or something similar, can be made to work as it is mostly about integers rather than reals.
discr1 , discr | none | The set.mm proof uses real number trichotomy.
sqrecd | ~ sqrecapd
expclzd | ~ expclzapd
exp0d | ~ expap0d
expne0d | ~ expap0d
expnegd | ~ expnegapd
exprecd | ~ exprecapd
expp1zd | ~ expp1zapd
expm1d | ~ expm1apd
expsubd | ~ expsubapd
sqdivd | ~ sqdivapd
expdivd | ~ expdivapd
reexpclzd | ~ reexpclzapd
sqgt0d | ~ sqgt0apd
mulsubdivbinom2 | none | Presumably provable if not equal is changed to apart.
muldivbinom2 | none | Presumably provable if not equal is changed to apart.
nn0le2msqi | ~ nn0le2msqd | Although nn0le2msqi could be proved, having a version in deduction form will be more useful.
nn0opthlem1 | ~ nn0opthlem1d | Although nn0opthlem1 could be proved, having a version in deduction form will be more useful.
nn0opthlem2 | ~ nn0opthlem2d | Although nn0opthlem2 could be proved, having a version in deduction form will be more useful.
nn0opthi | ~ nn0opthd | Although nn0opthi could be proved, having a version in deduction form will be more useful.
nn0opth2i | ~ nn0opth2d | Although nn0opth2i could be proved, having a version in deduction form will be more useful.
facmapnn | ~ faccl | We could express this in terms of ~ wf notation, or alternately define the mapping operation "^m" in iset.mm. But ~ faccl would be sufficient for the uses in set.mm.
faclbnd4 , faclbnd5 , and lemmas | none | Presumably provable; unused in set.mm.
bcval5 | ~ ibcval5
df-hash | ~ df-size
hashkf , hashgval , hashginv | none | Due to the differences between df-hash in set.mm and ~ df-size here, there's no particular need for these as stated
hashinf | ~ sizeinf
hashbnd | none | The set.mm proof is not intuitionistic.
hashfxnn0 , hashf , hashxnn0 , hashresfn , dmhashres , hashnn0pnf | none | Although ~ df-size is defined for finite sets and infinite sets, it is not clear we would be able to show this definition (or another definition) is defined for all sets.
hashnnn0genn0 | none | Not yet known whether this is provable or whether it is the sort of reverse closure theorem that we (at least so far) have been unable to intuitionize.
hashnemnf | none | Presumably provable but the set.mm proof relies on hashnn0pnf
hashv01gt1 | ~ sizefiv01gt1
hashfz1 | ~ sizefz1
hashen | ~ sizeen
hasheni | ~ sizeen , ~ sizeinf | It is not clear there would be any way to combine the finite and infinite cases.
hasheqf1o | ~ sizeeqf1o
hasheqf1oi | ~ sizeeqf1oi | It is not clear there would be any way to combine the finite and infinite cases.
hashf1rn | ~ sizef1rn | It is not clear there would be any way to combine the finite and infinite cases.
hasheqf1od | ~ sizeeqf1od
hashcard | none | Cardinality is not well developed in iset.mm
hashcl | ~ sizecl
hashxrcl | none | It is not clear there would be any way to combine the finite and infinite cases.
hashclb | none | Not yet known whether this is provable or whether it is the sort of reverse closure theorem that we (at least so far) have been unable to intuitionize.
nfile | ~ filtinf | It is not clear there would be any way to combine the case where ` A ` is finite and the case where it is infinite.
hashvnfin | none | This is a form of reverse closure, presumably not provable.
hashnfinnn0 | ~ sizeinf
isfinite4 | ~ isfinite4im
hasheq0 | ~ sizeeq0
hashneq0 , hashgt0n0 | ~ sizeneq0
hashnncl | ~ sizenncl
hash0 | ~ size0
hashsng | ~ sizesng
hashen1 | ~ sizeen1
hashrabrsn | none | Presumably would need conditions around the existence of ` A ` and decidability of ` ph ` but unused in set.mm.
hashrabsn01 | none | Presumably would need conditions around the existence of ` A ` and decidability of ` ph ` but unused in set.mm.
hashrabsn1 | none | The set.mm proof uses excluded middle and this theorem is unused in set.mm.
hashfn | ~ sizefn | There is an added condition that the domain be finite.
fseq1hash | ~ fseq1size
hashgadd | ~ omgadd
hashgval2 | none | Presumably provable, when reestated as ` ( size |`` _om ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` , but lightly used in set.mm.
hashdom | ~ sizedom | There is an added condition that ` B ` is finite.
hashdomi | ~ sizedom | It is presumably not possible to extend ~ sizedom beyond the finite set case.
hashun | ~ sizeun
hashun2 | none | The set.mm proof relies on undif2 (we just have ~ undif2ss ) and diffi (we just have ~ diffifi )
hashun3 | none | The set.mm proof relies on various theorems we do not have
hashinfxadd | none
hashunx | none | It is not clear there would be any way to combine the finite and infinite cases.
hashge0 | ~ sizecl | It is not clear there would be any way to combine the finite and infinite cases.
hashgt0 , hashge1 | ~ sizenncl | It is not clear there would be any way to combine the finite and infinite cases.
1elfz0hash | ~ 1elfz0size
hashnn0n0nn | ~ sizenncl | To the extent this is reverse closure, we probably can't prove it. For inhabited versus non-empty, see ~ fin0
hashunsng | ~ sizeunsng
hashprg | ~ sizeprg
elprchashprn2 | ~ sizesng | Given either ` N e. _V ` or ` -. N e. _V ` this could be proved (as ` ( size `` { M , N } ) ` reduces to ~ sizesng or ~ size0 respectively), but is not clear we can combine the cases (even ~ 1domsn may not be enough).
hashprb | ~ sizeprg
hashprdifel | none | This would appear to be a form of reverse closure.
prhash2ex | ~ prsize2ex
hashle00 | ~ sizeeq0
hashgt0elex , hashgt0elexb | ~ sizeneq0 | See ~ fin0 for inhabited versus non-empty. It isn't clear it would be possible to also include the infinite case as hashgt0elex does.
hashp1i | ~ sizep1i
hash1 , hash2 , hash3 , hash4 | ~ size1 , ~ size2 , ~ size3 , ~ size4
pr0hash2ex | ~ pr0size2ex
seqshft | none currently | need to figure out how to adjust this for the differences between set.mm and iset.mm concerning ` seq `.
df-sgn and theorems related to the sgn function | none | To choose a value near zero requires knowing the argument with unlimited precision. It would be possible to define for rational numbers, or real numbers apart from zero.
mulre | ~ mulreap
rediv | ~ redivap
imdiv | ~ imdivap
cjdiv | ~ cjdivap
sqeqd | none | The set.mm proof is not intuitionistic, and this theorem is unused in set.mm.
cjdivi | ~ cjdivapi
cjdivd | ~ cjdivapd
redivd | ~ redivapd
imdivd | ~ imdivapd
df-sqrt | ~ df-rsqrt | See discussion of complex square roots in the comment of ~ df-rsqrt . Here's one possibility if we do want to define square roots on (some) complex numbers: It should be possible to define the complex square root function on all complex numbers satisfying ` ( Im `` x ) # 0 \/ 0 <_ ( Re `` x ) ` , using a similar construction to the one used in set.mm. You need the real square root as a basis for the construction, but then there is a trick using the complex number x + |x| (see sqreu) that yields the complex square root whenever it is apart from zero (you need to divide by it at one point IIRC), which is exactly on the negative real line. You can either live with this constraint, which gives you the complex square root except on the negative real line (which puts a hole at zero), or you can extend it by continuity to zero as well by joining it with the real square root. The disjunctive domain of the resulting function might not be so useful though.
sqrtval | ~ sqrtrval | See discussion of complex square roots in the comment of ~ df-rsqrt
01sqrex and its lemmas | ~ resqrex | Both set.mm and iset.mm prove resqrex although via different mechanisms so there is no need for 01sqrex.
cnpart | none | See discussion of complex square roots in the comment of ~ df-rsqrt
sqrmo | ~ rsqrmo | See discussion of complex square roots in the comment of ~ df-rsqrt
resqreu | ~ rersqreu | Although the set.mm theorem is primarily about real square roots, the iset.mm equivalent removes some complex number related parts.
sqrtneg , sqrtnegd | none | Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of ~ df-rsqrt
sqrtm1 | none | Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of ~ df-rsqrt
absrpcl , absrpcld | ~ absrpclap , ~ absrpclapd
absdiv , absdivzi , absdivd | ~ absdivap , ~ absdivapzi , ~ absdivapd
absor , absori , absord | ~ qabsor | It also would be possible to prove this for real numbers apart from zero, if we wanted
absmod0 | none | See df-mod ; we may want to supply this for rationals or integers
absexpz | ~ absexpzap
max0add | ~ max0addsup
absz | none | Although this is presumably provable, the set.mm proof is not intuitionistic and it is lightly used in set.mm
recval | ~ recvalap
absgt0 , absgt0i | ~ absgt0ap , absgt0api
absmax | ~ maxabs
abs1m | none | Because this theorem provides ` ( * `` A ) / ( abs `` A ) ` as the answer if ` A =/= 0 ` and ` i ` as the answer if ` A = 0 ` , and uses excluded middle to combine those cases, it is presumably not provable as stated. We could prove the theorem with the additional condition that ` A # 0 ` , but it is unused in set.mm.
abslem2 | none | Although this could presumably be proved if not equal were changed to apart, it is lightly used in set.mm.
rddif , absrdbnd | none | If there is a need, we could prove these for rationals or real numbers apart from any rational. Alternately, we could prove a result with a slightly larger bound for any real number.
rexuzre | none | Unless the real number ` j ` is known to be apart from an integer, it isn't clear there would be any way to prove this (see the steps in the set.mm proof which rely on the floor of a real number). It is unused in set.mm for whatever that is worth.
caubnd | none | If we can prove fimaxre3 it would appear that the set.mm proof would work with small changes (in the case of the maximum of two real numbers, using ~ maxle1 , ~ maxle2 , and ~ maxcl ).
sqreulem , sqreu , sqrtcl , sqrtcld , sqrtthlem , sqrtf , sqrtth , sqsqrtd , msqsqrtd , sqr00d , sqrtrege0 , sqrtrege0d , eqsqrtor , eqsqrtd , eqsqrt2d | ~ rersqreu , ~ resqrtcl , ~ resqrtcld , ~ resqrtth | As described at ~ df-rsqrt , square roots of complex numbers are in set.mm defined with the help of excluded middle.
df-limsup and all superior limit theorems | none | This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-rlim and theorems related to limits of partial functions on the reals | none | This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to convergence.
df-o1 and theorems related to eventually bounded functions | none | This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-lo1 and theorems related to eventually upper bounded functions | none | This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
serclim0 | ~ iserclim0 | The only difference is the syntax of ` seq ` .
reccn2 | none yet | Will need to be revamped to deal with negated equality versus apartness and perhaps other issues.
clim2ser | ~ clim2iser | The only difference is the syntax of ` seq ` .
clim2ser2 | ~ clim2iser2 | The only difference is the syntax of ` seq ` .
iserex | ~ iiserex | The only difference is the syntax of ` seq ` .
isermulc2 | ~ iisermulc2 | The only difference is the syntax of ` seq ` .
iserle | ~ iserile | The only difference is the syntax of ` seq ` .
iserge0 | ~ iserige0 | The only difference is the syntax of ` seq ` .
climserle | ~ climserile | The only difference is the syntax of ` seq ` .
isershft | none yet | Relies on seqshft
isercoll and its lemmas, isercoll2 | none yet | The set.mm proof would need modification
climsup | none | To show convergence would presumably require a hypothesis related to the rate of convergence.
climbdd | none | Presumably could be proved but the current proof of caubnd would need at least some minor adjustments.
caurcvg2 | ~ climrecvg1n
caucvg | ~ climcvg1n
caucvgb | ~ climcaucn , ~ climcvg1n | Without excluded middle, there are additional complications related to the rate of convergence.
serf0 | ~ serif0 | The only difference is the syntax of ` seq ` .
iseralt | none | The set.mm proof relies on caurcvg2 which does not specify a rate of convergence.
df-sum | ~ df-isum | Although this defintion is intended to function similarly to the set.mm one, a lot of details have to be changed, especially around decidability, to make sum work.
sumex | none | This will need to replaced by suitable closure theorems.
sumeq2w | sumeq2 | Presumably could be proved, and perhaps also would rely only on extensionality (and logical axioms). But unused in set.mm.
sumeq2ii | ~ sumeq2d
df-prod and theorems using it | none | To define this, will need to tackle all the issues with ~ df-isum plus some more around, for example, not equal to zero versus apart from zero
znnen | none | Corollary 8.1.23 of [AczelRathjen] and thus presumably provable. The set.mm proof would not work as-is or with small changes, however.
qnnen | none | Corollary 8.1.23 of [AczelRathjen] and thus presumably provable. The set.mm proof would not work as-is or with small changes, however.
dvdsaddre2b | none | Something along these lines (perhaps with real changed to rational) may be possible
fsumdvds , 3dvds | none | May be possible when summation is well enough developed
sumeven , sumodd , evensumodd , oddsumodd , pwp1fsum , oddpwp1fsum | none | Presumably possible when summation is well enough developed
divalglem0 and other ~ divalg lemmas | ~ divalglemnn and other lemmas | Since the end result ~ divalg is the same, we don't list all the differences in lemmas here.
gcdcllem1 , gcdcllem2 , gcdcllem3 | ~ gcdn0cl , ~ gcddvds , ~ dvdslegcd | These are lemmas which are part of the proof of theorems that iset.mm proves a somewhat different way
seq1st | none | The second argument to ` seq ` , at least as handled in theorems such as ~ iseqfn , must be defined on all integers greater than or equal to ` M ` , not just at ` M ` itself. It may be possible to patch this up, but seq1st is unused in set.mm.
algr0 | ~ ialgr0 | Several hypotheses are tweaked or added to reflect differences in how ` seq ` works
algrf | ~ ialgrf | Several hypotheses are tweaked or added to reflect differences in how ` seq ` works
algrp1 | ~ ialgrp1 | Several hypotheses are tweaked or added to reflect differences in how ` seq ` works
alginv | ~ ialginv | Two hypotheses are tweaked or added to reflect differences in how ` seq ` works
algcvg | ~ ialgcvg | ` seq ` as defined in ~ df-iseq has a third argument
algcvga | ~ ialgcvga | Two hypotheses are tweaked or added to reflect differences in how ` seq ` works
algfx | ~ ialgfx | Two hypotheses are tweaked or added to reflect differences in how ` seq ` works
eucalg | ~ eucialg | Tweaked for the different ` seq ` syntax
df-lcmf and theorems using it | none | Although this could be defined, most of the theorems would need decidability conditions analogous to ~ zsupcl
absproddvds , absprodnn | none | Needs product to be developed, but once that is done seems like it might be possible.
fissn0dvds , fissn0dvdsn0 | none | Possibly could be proved using ~ findcard2 or the like.
coprmprod , coprmproddvds | none | Can investigate once product is better developed.
isprm5 | none | Presumably provable, but the set.mm proof relies on excluded middle in multiple places.
isprm7 | none | The set.mm proof relies on isprm5
maxprmfct | none | Presumably provable with suitable adjustments to the condition for the existence of the supremum
ncoprmlnprm | none | Presumably provable but the set.mm proof uses excluded middle

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