P. 93 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

redivapd 9201

Real part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 15-Jun-2020.)

⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℂ)    &   ⊢ (φ → A # 0)    ⇒   ⊢ (φ → (ℜ‘(B / A)) = ((ℜ‘B) / A))

Theorem

imdivapd 9202

Imaginary part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 15-Jun-2020.)

⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℂ)    &   ⊢ (φ → A # 0)    ⇒   ⊢ (φ → (ℑ‘(B / A)) = ((ℑ‘B) / A))

Theorem

crred 9203

The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    ⇒   ⊢ (φ → (ℜ‘(A + (i · B))) = A)

Theorem

crimd 9204

The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    ⇒   ⊢ (φ → (ℑ‘(A + (i · B))) = B)

3.6.2  Square root; absolute value

Syntax

csqrt 9205

Extend class notation to include square root of a complex number.

class √

Syntax

cabs 9206

Extend class notation to include a function for the absolute value (modulus) of a complex number.

class abs

Definition

df-rsqrt 9207*

Define a function whose value is the square root of a nonnegative real number.

Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root.

(Contributed by Jim Kingdon, 23-Aug-2020.)

⊢ √ = (x ∈ ℝ ↦ (℩y ∈ ℝ ((y↑2) = x ∧ 0 ≤ y)))

Definition

df-abs 9208

Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)

⊢ abs = (x ∈ ℂ ↦ (√‘(x · (∗‘x))))

Theorem

sqrtrval 9209*

Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)

⊢ (A ∈ ℝ → (√‘A) = (℩x ∈ ℝ ((x↑2) = A ∧ 0 ≤ x)))

Theorem

absval 9210

The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

⊢ (A ∈ ℂ → (abs‘A) = (√‘(A · (∗‘A))))

Theorem

rennim 9211

A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

⊢ (A ∈ ℝ → (i · A) ∉ ℝ⁺)

Theorem

sqrt0rlem 9212

Lemma for sqrt0 9213. (Contributed by Jim Kingdon, 26-Aug-2020.)

⊢ ((A ∈ ℝ ∧ ((A↑2) = 0 ∧ 0 ≤ A)) ↔ A = 0)

Theorem

sqrt0 9213

Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

⊢ (√‘0) = 0

Theorem

sqrtsq 9214

Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (√‘(A↑2)) = A)

Theorem

sqrtmsq 9215

Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (√‘(A · A)) = A)

Theorem

sqrt1 9216

The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

⊢ (√‘1) = 1

Theorem

sqrt4 9217

The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

⊢ (√‘4) = 2

Theorem

sqrt9 9218

The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

⊢ (√‘9) = 3

Theorem

absneg 9219

Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

⊢ (A ∈ ℂ → (abs‘-A) = (abs‘A))

Theorem

abscj 9220

The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)

⊢ (A ∈ ℂ → (abs‘(∗‘A)) = (abs‘A))

Theorem

absval2 9221

Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

⊢ (A ∈ ℂ → (abs‘A) = (√‘(((ℜ‘A)↑2) + ((ℑ‘A)↑2))))

Theorem

abs0 9222

The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

⊢ (abs‘0) = 0

Theorem

absi 9223

The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

⊢ (abs‘i) = 1

Theorem

abs00bd 9224

If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)

⊢ (φ → A = 0)    ⇒   ⊢ (φ → (abs‘A) = 0)

Theorem

absid 9225

A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (abs‘A) = A)

Theorem

abs1 9226

The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

⊢ (abs‘1) = 1

Theorem

absnid 9227

A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

⊢ ((A ∈ ℝ ∧ A ≤ 0) → (abs‘A) = -A)

Theorem

absre 9228

Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

⊢ (A ∈ ℝ → (abs‘A) = (√‘(A↑2)))

Theorem

nn0abscl 9229

The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)

⊢ (A ∈ ℤ → (abs‘A) ∈ ℕ₀)

Theorem

zabscl 9230

The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

⊢ (A ∈ ℤ → (abs‘A) ∈ ℤ)

PART 4  GUIDES AND MISCELLANEA

4.1  Guides (conventions, explanations, and examples)

4.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 9231

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (aka "set.mm"). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in metamath. You can find some development using CZF in BJ's mathbox starting at ax-bd0 9268 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5454, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). For this reason, we generally need to prove we are evaluating functions within their domains and avoid the reverse closure theorems of the Metamath Proof Explorer.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

Abbreviation	Mnenomic	Source	Expression	Syntax?	Example(s)
ap	apart	df-ap 7366		Yes	apadd1 7392, apne 7407

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/forum/#!forum/metamath.

(Contributed by Jim Kingdon, 24-Feb-2020.)

⊢ φ    ⇒   ⊢ φ

PART 5  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

5.1  Mathboxes for user contributions

5.1.1  Mathbox guidelines

Theorem

mathbox 9232

(This theorem is a dummy placeholder for these guidelines. The name of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only 0-ary class constants for new definitions.

2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. We may decide to move some theorems to the main part of set.mm for general use.

2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

⊢ x = x

5.2  Mathbox for Mykola Mostovenko

Theorem

ax1hfs 9233

Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)

⊢ (φ → (φ ∧ φ))

5.3  Mathbox for BJ

5.3.1  Propositional calculus

Theorem

nnexmid 9234

Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ (φ ∨ ¬ φ)

Theorem

nndc 9235

Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ DECID φ

Theorem

dcdc 9236

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (DECID DECID φ ↔ DECID φ)

5.3.2  Predicate calculus

Theorem

bj-ex 9237*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1486 and 19.9ht 1529 or 19.23ht 1383). (Proof modification is discouraged.)

⊢ (∃xφ → φ)

Theorem

bj-hbalt 9238

Closed form of hbal 1363 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀y(φ → ∀xφ) → (∀yφ → ∀x∀yφ))

Theorem

bj-nfalt 9239

Closed form of nfal 1465 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀xℲyφ → Ⅎy∀xφ)

Theorem

spimd 9240

Deduction form of spim 1623. (Contributed by BJ, 17-Oct-2019.)

⊢ (φ → Ⅎxχ)    &   ⊢ (φ → ∀x(x = y → (ψ → χ)))    ⇒   ⊢ (φ → (∀xψ → χ))

Theorem

2spim 9241*

Double substitution, as in spim 1623. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎxχ    &   ⊢ Ⅎzχ    &   ⊢ ((x = y ∧ z = 𝑡) → (ψ → χ))    ⇒   ⊢ (∀z∀xψ → χ)

Theorem

ch2var 9242*

Implicit substitution of y for x and 𝑡 for z into a theorem. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎxψ    &   ⊢ Ⅎzψ    &   ⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ))    &   ⊢ φ    ⇒   ⊢ ψ

Theorem

ch2varv 9243*

Version of ch2var 9242 with non-freeness hypotheses replaced by DV conditions. (Contributed by BJ, 17-Oct-2019.)

⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ))    &   ⊢ φ    ⇒   ⊢ ψ

Theorem

bj-exlimmp 9244

Lemma for bj-vtoclgf 9250. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎxψ    &   ⊢ (χ → φ)    ⇒   ⊢ (∀x(χ → (φ → ψ)) → (∃xχ → ψ))

Theorem

bj-exlimmpi 9245

Lemma for bj-vtoclgf 9250. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎxψ    &   ⊢ (χ → φ)    &   ⊢ (χ → (φ → ψ))    ⇒   ⊢ (∃xχ → ψ)

Theorem

bj-sbimedh 9246

A strengthening of sbiedh 1667 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (φ → ∀xφ)    &   ⊢ (φ → (χ → ∀xχ))    &   ⊢ (φ → (x = y → (ψ → χ)))    ⇒   ⊢ (φ → ([y / x]ψ → χ))

Theorem

bj-sbimeh 9247

A strengthening of sbieh 1670 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (ψ → ∀xψ)    &   ⊢ (x = y → (φ → ψ))    ⇒   ⊢ ([y / x]φ → ψ)

Theorem

bj-sbime 9248

A strengthening of sbie 1671 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ Ⅎxψ    &   ⊢ (x = y → (φ → ψ))    ⇒   ⊢ ([y / x]φ → ψ)

5.3.3  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 9249

Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    &   ⊢ (x = A → φ)    ⇒   ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ 𝑉 → ψ))

Theorem

bj-vtoclgf 9250

Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    &   ⊢ (x = A → φ)    &   ⊢ (x = A → (φ → ψ))    ⇒   ⊢ (A ∈ 𝑉 → ψ)

Theorem

elabgf0 9251

Lemma for elabgf 2679. (Contributed by BJ, 21-Nov-2019.)

⊢ (x = A → (A ∈ {x ∣ φ} ↔ φ))

Theorem

elabgft1 9252

One implication of elabgf 2679, in closed form. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    ⇒   ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ {x ∣ φ} → ψ))

Theorem

elabgf1 9253

One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    &   ⊢ (x = A → (φ → ψ))    ⇒   ⊢ (A ∈ {x ∣ φ} → ψ)

Theorem

elabgf2 9254

One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    &   ⊢ (x = A → (ψ → φ))    ⇒   ⊢ (A ∈ B → (ψ → A ∈ {x ∣ φ}))

Theorem

elabf1 9255*

One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎxψ    &   ⊢ (x = A → (φ → ψ))    ⇒   ⊢ (A ∈ {x ∣ φ} → ψ)

Theorem

elabf2 9256*

One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎxψ    &   ⊢ A ∈ V    &   ⊢ (x = A → (ψ → φ))    ⇒   ⊢ (ψ → A ∈ {x ∣ φ})

Theorem

elab1 9257*

One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.)

⊢ (x = A → (φ → ψ))    ⇒   ⊢ (A ∈ {x ∣ φ} → ψ)

Theorem

elab2a 9258*

One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.)

⊢ A ∈ V    &   ⊢ (x = A → (ψ → φ))    ⇒   ⊢ (ψ → A ∈ {x ∣ φ})

Theorem

elabg2 9259*

One implication of elabg 2682. (Contributed by BJ, 21-Nov-2019.)

⊢ (x = A → (ψ → φ))    ⇒   ⊢ (A ∈ 𝑉 → (ψ → A ∈ {x ∣ φ}))

Theorem

bj-rspgt 9260

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ ℲxB    &   ⊢ Ⅎxψ    ⇒   ⊢ (∀x(x = A → (φ → ψ)) → (∀x ∈ B φ → (A ∈ B → ψ)))

Theorem

bj-rspg 9261

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ ℲxA    &   ⊢ ℲxB    &   ⊢ Ⅎxψ    &   ⊢ (x = A → (φ → ψ))    ⇒   ⊢ (∀x ∈ B φ → (A ∈ B → ψ))

Theorem

cbvrald 9262*

Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

⊢ Ⅎxφ    &   ⊢ Ⅎyφ    &   ⊢ (φ → Ⅎyψ)    &   ⊢ (φ → Ⅎxχ)    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀y ∈ A χ))

Theorem

bj-intabssel 9263

Version of intss1 3621 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ ℲxA    ⇒   ⊢ (A ∈ 𝑉 → ([A / x]φ → ∩ {x ∣ φ} ⊆ A))

Theorem

bj-intabssel1 9264

Version of intss1 3621 using a class abstraction and implicit substitution. Closed form of intmin3 3633. (Contributed by BJ, 29-Nov-2019.)

⊢ ℲxA    &   ⊢ Ⅎxψ    &   ⊢ (x = A → (ψ → φ))    ⇒   ⊢ (A ∈ 𝑉 → (ψ → ∩ {x ∣ φ} ⊆ A))

Theorem

bj-elssuniab 9265

Version of elssuni 3599 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ ℲxA    ⇒   ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ}))

Theorem

bj-sseq 9266

If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)

⊢ (φ → (ψ ↔ A ⊆ B))    &   ⊢ (φ → (χ ↔ B ⊆ A))    ⇒   ⊢ (φ → ((ψ ∧ χ) ↔ A = B))

5.3.4  Bounded formulas

This is an ongoing project to define bounded formulas, following a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein).

In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ₀) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ₀) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitonistic, for instance to state the axiom scheme of Δ₀-induction.

To formalize this in Metamath, there are several choices to make.

A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph₀ ...) and an axiom "$a wff ph₀ " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph₀ -> ps₀ )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED φ " is a formula meaning that φ is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.)

A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, ∀x ⊤ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊤ which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded.

A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 9268. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (φ → BOUNDED φ) and ⊢ (¬ φ → BOUNDED φ) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.)

Having ax-bd0 9268 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 9269 through ax-bdsb 9277) can be written either in closed or inference form. The fact that ax-bd0 9268 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness.

Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that x ∈ 𝜔 is a bounded formula. However, since 𝜔 can be defined as "the y such that PHI" a proof using the fact that x ∈ 𝜔 is bounded can be converted to a proof in iset.mm by replacing 𝜔 with y everywhere and prepending the antecedent PHI, since x ∈ y is bounded by ax-bdel 9276. For a similar method, see bj-omtrans 9416.

Note that one cannot add an axiom ⊢ BOUNDED x ∈ A since by bdph 9305 it would imply that every formula is bounded.

For CZF, a useful set of notes is

Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf)

and an interesting article is

Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204)

Syntax

wbd 9267

Syntax for the predicate BOUNDED.

wff BOUNDED φ

Axiom

ax-bd0 9268

If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)

⊢ (φ ↔ ψ)    ⇒   ⊢ (BOUNDED φ → BOUNDED ψ)

Axiom

ax-bdim 9269

An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    ⇒   ⊢ BOUNDED (φ → ψ)

Axiom

ax-bdan 9270

The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    ⇒   ⊢ BOUNDED (φ ∧ ψ)

Axiom

ax-bdor 9271

The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    ⇒   ⊢ BOUNDED (φ ∨ ψ)

Axiom

ax-bdn 9272

The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED ¬ φ

Axiom

ax-bdal 9273*

A bounded universal quantification of a bounded formula is bounded. Note the DV condition on x, y. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED ∀x ∈ y φ

Axiom

ax-bdex 9274*

A bounded existential quantification of a bounded formula is bounded. Note the DV condition on x, y. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED ∃x ∈ y φ

Axiom

ax-bdeq 9275

An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED x = y

Axiom

ax-bdel 9276

An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED x ∈ y

Axiom

ax-bdsb 9277

A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1643, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED [y / x]φ

Theorem

bdeq 9278

Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

⊢ (φ ↔ ψ)    ⇒   ⊢ (BOUNDED φ ↔ BOUNDED ψ)

Theorem

bd0 9279

A formula equivalent to a bounded one is bounded. See also bd0r 9280. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ (φ ↔ ψ)    ⇒   ⊢ BOUNDED ψ

Theorem

bd0r 9280

A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9279) biconditional in the hypothesis, to work better with definitions (ψ is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ (ψ ↔ φ)    ⇒   ⊢ BOUNDED ψ

Theorem

bdbi 9281

A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    ⇒   ⊢ BOUNDED (φ ↔ ψ)

Theorem

bdstab 9282

Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED STAB φ

Theorem

bddc 9283

Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED DECID φ

Theorem

bd3or 9284

A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    &   ⊢ BOUNDED χ    ⇒   ⊢ BOUNDED (φ ∨ ψ ∨ χ)

Theorem

bd3an 9285

A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    &   ⊢ BOUNDED χ    ⇒   ⊢ BOUNDED (φ ∧ ψ ∧ χ)

Theorem

bdth 9286

A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

⊢ φ    ⇒   ⊢ BOUNDED φ

Theorem

bdtru 9287

The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED ⊤

Theorem

bdfal 9288

The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED ⊥

Theorem

bdnth 9289

A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

⊢ ¬ φ    ⇒   ⊢ BOUNDED φ

Theorem

bdnthALT 9290

Alternate proof of bdnth 9289 not using bdfal 9288. Then, bdfal 9288 can be proved from this theorem, using fal 1249. The total number of proof steps would be 17 (for bdnthALT 9290) + 3 = 20, which is more than 8 (for bdfal 9288) + 9 (for bdnth 9289) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ¬ φ    ⇒   ⊢ BOUNDED φ

Theorem

bdxor 9291

The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    &   ⊢ BOUNDED ψ    ⇒   ⊢ BOUNDED (φ ⊻ ψ)

Theorem

bj-bdcel 9292*

Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)

⊢ BOUNDED y = A    ⇒   ⊢ BOUNDED A ∈ x

Theorem

bdab 9293

Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED x ∈ {y ∣ φ}

Theorem

bdcdeq 9294

Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

⊢ BOUNDED φ    ⇒   ⊢ BOUNDED CondEq(x = y → φ)

5.3.5  Bounded classes

In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 9296. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas.

As will be clear by the end of this subsection (see for instance bdop 9330), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED φ ⇒ ⊢ BOUNDED ⟨{x ∣ φ}, ({y, suc z} × ⟨𝑡, ∅⟩)⟩. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like ⊢ BOUNDED A ⇒ ⊢ BOUNDED {A}.

Syntax

wbdc 9295

Syntax for the predicate BOUNDED.

wff BOUNDED A

Definition

df-bdc 9296*

Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

⊢ (BOUNDED A ↔ ∀xBOUNDED x ∈ A)

Theorem

bdceq 9297

Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

⊢ A = B    ⇒   ⊢ (BOUNDED A ↔ BOUNDED B)

Theorem

bdceqi 9298

A class equal to a bounded one is bounded. Note the use of ax-ext 2019. See also bdceqir 9299. (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED A    &   ⊢ A = B    ⇒   ⊢ BOUNDED B

Theorem

bdceqir 9299

A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9298) equality in the hypothesis, to work better with definitions (B is the definiendum that one wants to prove bounded; see comment of bd0r 9280). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED A    &   ⊢ B = A    ⇒   ⊢ BOUNDED B

Theorem

bdel 9300*

The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

⊢ (BOUNDED A → BOUNDED x ∈ A)