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Created by Mario Carneiro

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) → (𝜓𝜑)) (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 → (ψφ))
    Axiom Frege ax-2 ((φ → (ψχ)) → ((φψ) → (φχ)))
    Rule of Modus Ponens ax-mp ` |- ph `   &   ` |- ph -> ps `   ⇒   ` |- ps `
    Left 'and' eliminationax-ia1 ` |- ( ( ph /\ ps ) -> ph ) `
    Right 'and' eliminationax-ia2 ((φ ψ) → ψ)
    'And' introductionax-ia3 (φ → (ψ → (φ ψ)))
    Definition of 'or'ax-io (((φ χ) → ψ) ↔ ((φψ) (χψ)))
    'Not' introductionax-in1 ((φ → ¬ φ) → ¬ φ)
    'Not' eliminationax-in2 φ → (φψ))

    Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φψ is not equivalent to ¬ φψ, nor is φψ equivalent to ¬ φψ, nor is φψ 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 (xφφ)
    Axiom of Quantified Implication ax-5 (x(φψ) → (xφxψ))
    The converse of ax-5o ax-i5r ((xφxψ) → x(xφψ))
    Axiom of Quantifier Commutation ax-7 (xyφyxφ)
    Rule of Generalization ax-gen φ   =>    xφ
    x is bound in xφ ax-ial (xφxxφ)
    x is bound in xφ ax-ie1 (xφxxφ)
    Define existential quantification ax-ie2 (x(ψxψ) → (x(φψ) ↔ (xφψ)))
    Axiom of Equality ax-8 (x = y → (x = zy = z))
    Axiom of Existence ax-i9 x x = y
    Axiom of Quantifier Substitution ax-10 (x x = yy y = x)
    Axiom of Variable Substitution ax-11 (x = y → (yφx(x = yφ)))
    Axiom of Quantifier Introduction ax-i12 (z z = x (z z = y z(x = yz x = y)))
    Axiom of Bundling ax-bndl (z z = x (z z = y xz(x = yz x = y)))
    Left Membership Equality ax-13 (x = y → (x zy z))
    Right Membership Equality ax-14 (x = y → (z xz y))
    Distinctness ax-17 (φxφ), 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 e. 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 xz y) → x = y)
    Axiom of Collection ax-coll (x 𝑎 yφ𝑏x 𝑎 y 𝑏 φ)
    Axiom of Separation ax-sep yx(x y ↔ (x z φ))
    Axiom of Power Sets ax-pow yz(w(w zw x) → z y)
    Axiom of Pairing ax-pr zw((w = x w = y) → w z)
    Axiom of Union ax-un yz(w(z w w x) → z y)
    Axiom of Set Induction ax-setind (𝑎(y 𝑎 [y / 𝑎]φφ) → 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.


    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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