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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 (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) "" provides in Metamath a development of "" 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, adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of

    Among these new axioms, the 6 first (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 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 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-bnd (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-bnd 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 φ   &   φψ   =>   ψ
    Left 'and' eliminationax-ia1 ((φ ψ) → φ)
    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 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-bnd (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
  • this space as we build out more of set theory

  • 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. 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.
    notnotrd , notnotri , notnot2 , notnot notnot1 Double negation introduction holds but not double negation elimination.
    mt3d mtod
    nyl2 nsyl
    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
    xor3 , nbbn xorbin, xornbi, xor3dc, nbbndc
    biass biassdc
    df-or , pm4.64 , pm2.54 , orri , orrd pm2.53, ori, ord
    pm4.63 pm3.2im
    iman imanim
    annim annimim
    oran oranim
    ianor pm3.14
    biluk bilukdc
    ecase3d none This is a form of case elimination.
    dedlem0b dedlemb
    3ianor 3ianorr
    df-ex exalim
    alex alexim
    exnal exnalim
    alexn alexnim
    exanali exanaliim
    19.35 19.35-1
    19.30 none
    19.36 19.36-1
    19.37 19.37-1
    19.32 19.32r
    19.31 19.31r
    exdistrf exdistrfor
    exmo exmonim
    mo2 mo2r, mo3
    nne none The reverse direction could be proved; the forward direction is double negation elimination.
    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
    ralcom2 ralcom
    sspss sspssr
    n0f n0rf
    n0 n0r
    neq0 neq0r
    reximdva0 reximdva0m
    r19.45zv r19.45mv
    dfif2 df-if
    ifsb none
    dfif4 none Unused in
    dfif5 none Unused in
    ifnot none
    ifan none
    ifor none
    ifeq1da, ifeq2da none
    ifclda none
    ifeqda none
    elimif , ifbothda , ifboth , ifid , eqif , ifval , elif , ifel , ifcl , ifcld , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD none
    dedth , dedth2h , dedth3h , dedth4h , dedth2v , dedth3v , dedth4v , elimhyp , elimhyp2v , elimhyp3v , elimhyp4v , elimel , elimdhyp , keephyp , keephyp2v , keephyp3v , keepel , ifex , ifexg none Even in, the weak deduction theorem is discouraged in favor of theorems in deduction form.
    trintss trintssm
    trint0 trint0m
    csbexg , csbex csbexga, csbexa uses case elimination to remove the A V condition.
    intabs none Lightly used in, and the proof is not intuitionistic
    moabex euabex In general, most of the ∃! theorems still hold, but a decent number of the ∃* ones get caught up on "there are two cases: the set exists or it does not"
    snex snexg, snex The version of snex has an additional hypothesis
    opex opexg, opex The version of opex has additional hypotheses
    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 𝜔
    pitric For the specific order <N Or N
    nqtric For the specific order <Q Or Q
    sotrieq sotritrieq For a trichotomous order
    sotrieq2 see sotrieq and then apply ioran
    issoi issod, ispod Many of the 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 and all theorems using Fr none Because assumes ax-setind without reluctance, all sets are well-founded. We could adopt a treatment more like if people want to investigate set theories which are constructive but which do not assume ax-setind.
    df-we (and all theorems using We) none Ordering is moderately different in constructive logic, so if there is anything along these lines worth doing it will be different from
    tz7.7 none
    ordelssne none
    ordelpss none
    ordsseleq onelss, eqimss Taken together, onelss and eqimss represent the reverse direction of the biconditional from ordsseleq
    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 , ordtri2or2 , 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 See also ordelpss in
    ordtr3 none This is weak linearity of ordinals, which presumably implies excluded middle by ordsoexmid.
    ordtri2or none Implies excluded middle as shown at ordtri2orexmid.
    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
    dmxpid dmxpm
    relimasn imasng
    opswap opswapg
    cnvso cnvsom
    iotaex euiotaex
    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
    1stexg, 2ndexg for the functions 1st and 2nd
    ndmfv ndmfvg The ¬ A V case is fvprc.
    elfvdm relelfvdm
    funiunfv fniunfv, funiunfvdm
    funiunfvf funiunfvdmf
    eluniima eluniimadm
    riotaex riotacl
    nfriotad nfriotadxy
    csbriota , csbriotagOLD csbriotag
    riotaxfrd none Although it may be intuitionizable, it is lightly used in
    ovex fnovex Sometimes or fvexg or relrnfvex may work.
    ov3 ovi3 Although's ov3 could be proved, it is only used a few places in, and in those places need the modified form found in ovi3.
    oprssdm none
    ndmovg , ndmov , ndmovcl , ndmovrcl , ndmovcom , ndmovass , ndmovdistr , ndmovord , and ndmovordi none Although these theorems are moderately widely used in, in many cases they are being used for case elimination and the proofs are not intuitionistic. We might bring back some of them later (which may have conditions that the sets exist, but aren't in the domain), but maybe we'll find a way to avoid them.
    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. A = On → A𝑉 is provable, and (B On AB) → A 𝑉 is provable. (Why isn't df-pss stated so that the set difference is inhabited? If so, you could prove A ⊊ On → A 𝑉.)
    onint none Conjectured to not be provable without excluded middle. If you apply onint to a pair you can derive totality of the order.
    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 none yet This one (with non-empty changed to inhabited) I think can still be salvaged though. From the fact that it is inhabited you get that it exists, and is a subset of an ordinal x. It is an intersection of transitive sets so it is transitive, and of course all its members are members of x so they are transitive too. And E. Fr A falls to subsets.
    onintrab, onintrab2 none yet The proof uses oninton.
    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 yet falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.
    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 ∩ 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
    onsucuni2 none yet Conjectured to be provable.
    0elsuc none yet Conjectured to be provable.
    onuniorsuci none Conjectured to not be provable without excluded middle.
    onuninsuci, ordununsuc none Conjectured to be provable in the forward direction but not the reverse one.
    onsucssi none yet Conjectured to be provable.
    nlimsucg none yet Conjectured to be provable.
    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 ∩ 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 {∅} ∣ φ} 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
    1stval 1stvalg
    2ndval 2ndvalg
    1stnpr none May be intuitionizable, but very lightly used in
    2ndnpr none May be intuitionizable, but very lightly used in
    brtpos brtposg
    ottpos ottposg
    ovtpos ovtposg
    pwuninel pwuninel2 The 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
    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, ondif1 is used for Cantor Normal Form
    ondif2 , dif20el none The proof is not intuitionistic
    brwitnlem none The 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 proof of omwordi relies on case elimination.
    omword1 nnmword
    nnawordex nnaordex nnawordex is only used a few places in
    swoso none Unused in
    ecdmn0 ecdmn0m
    erdisj, qsdisj, qsdisj2, uniinqs none These could presumably be restated to be provable, but they are lightly used in
    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
    eceqoveq none Unused in
    2pwuninel 2pwuninelg
    ax-reg , axreg2 , zfregcl ax-setind ax-reg implies excluded middle as seen at regexmid
    addcompi addcompig
    addasspi addasspig
    mulcompi mulcompig
    mulasspi mulasspig
    distrpi distrpig
    addcanpi addcanpig
    mulcanpi mulcanpig
    addnidpi addnidpig
    ltapi ltapig
    ltmpi ltmpig
    nlt1pi nlt1pig
    indpi finds Although indpi presumably could be proved, it is lightly used in
    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
    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 none The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle.
    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 × 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 proof of recexsr), but "not equal to zero" would need to be changed to "apart from zero".
    mappsrpr , ltpsrpr , map2psrpr none Although variants of these theorems could be intuitionized, in they are only used for supremum theorems, so we can consider this in more detail when we tackle what kind of supremum theorems to prove.
    supsrlem , supsr none The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle.
    axaddf , ax-addf , axmulf , ax-mulf none Because these are described as deprecated in, we haven't figured out what would be involved in proving them for
    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 none yet The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. If we want a set of axioms for real numbers which allows us to avoid construction-dependent theorems beyond this point, we'll need a modified Least Upper Bound property, a statement concerning Dedekind cuts or something similar, or some other axiom(s).
    elimne0 none Even in, the weak deduction theorem is discouraged in favor of theorems in deduction form.
    xrltnle xrlenlt
    ssxr df-xr Lightly used in
    ltnle , ltnlei , ltnled lenlt, zltnle
    lttri2 , lttri4 ztri3or Real number trichotomy is not provable.
    leloe , eqlelt , leloei , leloed , eqleltd none
    leltne , leltned none
    ltneOLD ltne
    letric , letrii , letrid zletric
    ltlen , ltleni , ltlend ltleap, zltlen
    ne0gt0 none We presumably could prove this if we changed "not equal to zero" to "apart from zero".
    lecasei , lelttric , ltlecasei none These are real number trichotomy
    lttri2i none This can be read as "two real numbers are non-equal if and only if they are apart" which relies on excluded middle
    ne0gt0d , lttrid , lttri2d , lttri4d none These are real number trichotomy
    dedekind , dedekindle none
    mul02lem1 none The one use in is not needed in
    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 These depend on real number trichotomy and are not used until later in
    wloglei , wlogle none These depend on real number trichotomy and are not used until later in
    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 none This probably could be proved in terms of tightness of apartness and A # 0 B # 0 → (A · B) # 0, but is unused in
    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 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
    divdir , divcan3 , divcan4 divdirap, divcanap3, divcanap4
    div11 , divid , div0 div11ap, dividap, div0ap
    diveq1 , divneg , divsubdir diveqap1, divnegap, 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
    rereccld , redivcld rerecclapd, redivclapd
    elimgt0 , elimge0 none Even in, the weak deduction theorem is discouraged in favor of theorems in deduction form.
    mulge0b , mulle0b , mulsuble0b none Presumably unprovable for reasons analogous to mul0or.
    ledivp1i , ltdivp1i none Presumably could be proved, but unused in
    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 none
    nnunb , arch , nnrecl , bndndx none Dealing with the Archimedean property will need to await adding some form of supremum (or other property of sets of reals). Our other real number axioms are first-order, and therefore are not sufficient to rule out non-Archimedean fields such as the hyperreal numbers.
    nn0n0n1ge2 , nn0n0n1ge2b none yet These should be provable, and might be easy now that we have ztri3or.
    frnnn0supp , frnnn0fsupp nn0supp 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.
    elz2 , dfz2 none yet Should be provable, but might need to wait until we have proved trichotomy of integers.
    suprzcl none
    btwnz none Depends on arch
    zriotaneg none Lightly used in
    suprfinzcl none
    decex deccl
    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 , max1 , max2 , min1 , min2 , maxle , lemin , maxlt , ltmin , max0sub , ifle none
    qbtwnre , qbtwnxr none yet Once we tackle the Archimedean principle, we should be able to prove these.
    qsqueeze none yet Once we have qbtwnre , we should be able to prove this as a consequence of squeeze0.
    qextltlem , qextlt , qextle none The proof is not intuitionistic.
    xralrple , alrple none yet If we had qbtwnxr , it looks like the 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, and the proofs would require significant changes.

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