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PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY

Logic can be defined as the "study of the principles of correct reasoning" (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in Informal Reasoning and Education ) or as "a formal system using symbolic techniques and mathematical methods to establish truth-values" (the Oxford English Dictionary).

This section formally defines the logic system we will use. In particular, it defines symbols for declaring truthful statements, along with rules for deriving truthful statements from other truthful statements. The system defined here is intuitionistic first-order logic with equality.

We begin with a few housekeeping items in pre-logic, and then introduce propositional calculus (both its axioms and important theorems that can be derived from them). Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. This is followed by proofs that other axiomatizations of classical propositional calculus can be derived from the axioms we have chosen to use.

We then define predicate calculus, which adds additional symbols and rules useful for discussing objects (beyond simply true or false). In particular, it introduces the symbols ("equals"), ("is a member of"), and ("for all"). The first two are called "predicates." A predicate specifies a true or false relationship between its two arguments.

1.1  Pre-logic

This section includes a few "housekeeping" mechanisms before we begin defining the basics of logic.

1.1.1  Inferences for assisting proof development

The inference rules in this section will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.

Theoremidi 1 (Note: This inference rule and the next one, a1ii 2, will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic; please start with the statement wn 3 instead.)

This inference says "if is true then is true". This inference requires no axioms for its proof, and is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant. You can remove this using the metamath-exe (Metamath program) Proof Assistant using the "MM-PA> MINIMIZE_WITH *" command. This is the inference associated with id 19, hence its name. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema1ii 2 (Note: This inference rule and the previous one, idi 1, will normally never appear in a completed proof.)

This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step.

The Metamath (Metamath-exe) program Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof.

Instructions: (1) Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since "MM-PA> ASSIGN LAST" can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a "MM-PA> LET STEP" command to pre-assign the conclusion of the independent subproof to a1ii.2. (3) After the independent subproof is complete, use "MM-PA> IMPROVE ALL" to assign it automatically to an unknown step in the main proof that matches it. (4) After the entire proof is complete, use "MM-PA> MINIMIZE_WITH *" to clean up (discard) all a1ii 2 references automatically.

This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with a1i 9. (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

1.2  Propositional calculus

Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. The simplest propositional truth is , which can be read "if something is true, then it is true" - rather trivial and obvious, but nonetheless it must be proved from the axioms (see theorem id 19).

Our system of propositional calculus consists of a few basic axioms and a unique rule of inference, modus ponens.

The propositional calculus used here is the intuitionistic propositional calculus.

All 194 axioms, definitions, and theorems for propositional calculus in Principia Mathematica (specifically *1.2 through *5.75) are axioms or formally proven. See the Bibliographic Cross-References at http://us.metamath.org/ileuni/mmbiblio.html 19 for a complete cross-reference from sources used to its formalization in the Intuitionistic Logic Explorer.

1.2.1  Recursively define primitive wffs for propositional calculus

Syntaxwn 3 If is a wff, so is or "not ." Part of the recursive definition of a wff (well-formed formula). Traditionally, Greek letters are used to represent wffs, and we follow this convention. In propositional calculus, we define only wffs built up from other wffs, i.e. there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition by including atomic wffs (weq 1479 and wel 1481).

Syntaxwi 4 If and are wff's, so is or " implies ." Part of the recursive definition of a wff. The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of , the middle may be informally called either an antecedent or part of the consequent depending on context.

1.2.2  Propositional logic axioms for implication

Axiomax-mp 5 Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if is true, and implies , then must also be true." This rule is sometimes called "detachment", since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens", a Latin phrase that means "the mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens mto 651.

Note: In some web page displays such as the Statement List, the symbols " " and " " informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies." They are not part of the formal language. (Contributed by NM, 30-Sep-1992.)

Axiomax-1 6 Axiom Simp. Axiom A1 of [Margaris] p. 49. One of the axioms of propositional calculus. This axiom is called Simp or "the principle of simplification" in Principia Mathematica (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of and to the assertion of simply."

The theorems of propositional calculus are also called tautologies. Although classical propositional logic tautologies can be proved using truth tables, there is no similarly simple system for intuitionistic propositional logic, so proving tautologies from axioms is the preferred approach. (Contributed by NM, 5-Aug-1993.)

Axiomax-2 7 Axiom Frege. Axiom A2 of [Margaris] p. 49. This axiom "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by pm5.41 250. (Contributed by NM, 5-Aug-1993.)

1.2.3  Logical implication

The results in this section are based on implication only, and only use ax-1 6, ax-2 7, and ax-mp 5. In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent".

We will use the following descriptive terms very loosely: A "closed form" or "tautology" has no \$e hypotheses. An "inference" has one or more \$e hypotheses. A "deduction" is an inference in which the hypotheses and the conclusion share the same antecedent.

Theoremmp2b 8 A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)

Theorema1i 9 Inference derived from axiom ax-1 6. See a1d 22 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 45. (Contributed by NM, 5-Aug-1993.)

Theoremmp1i 10 Drop and replace an antecedent. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Theorema2i 11 Inference derived from axiom ax-2 7. (Contributed by NM, 5-Aug-1993.)

Theoremimim2i 12 Inference adding common antecedents in an implication. (Contributed by NM, 5-Aug-1993.)

Theoremmpd 13 A modus ponens deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl 14 An inference version of the transitive laws for implication imim2 55 and imim1 76, which Russell and Whitehead call "the principle of the syllogism...because...the syllogism in Barbara is derived from them" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism." (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.)

Theoremmpi 15 A nested modus ponens inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)

Theoremmp2 16 A double modus ponens inference. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theorem3syl 17 Inference chaining two syllogisms. (Contributed by NM, 5-Aug-1993.)

Theorem4syl 18 Inference chaining three syllogisms. The use of this theorem is marked "discouraged" because it can cause the "minimize" command to have very long run times. However, feel free to use "minimize 4syl /override" if you wish. (Contributed by BJ, 14-Jul-2018.) (New usage is discouraged.)

Theoremid 19 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see idALT 20. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)

TheoremidALT 20 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 17 (PDF p. 23) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. For a shorter version of the proof that takes advantage of previously proved theorems, see id 19. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) Use id 19 instead. (New usage is discouraged.)

Theoremidd 21 Principle of identity with antecedent. (Contributed by NM, 26-Nov-1995.)

Theorema1d 22 Deduction introducing an embedded antecedent. (The proof was revised by Stefan Allan, 20-Mar-2006.)

Naming convention: We often call a theorem a "deduction" and suffix its label with "d" whenever the hypotheses and conclusion are each prefixed with the same antecedent. This allows us to use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used; here would be replaced with a conjunction (wa 103) of the hypotheses of the would-be deduction. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 9. Finally, a "theorem" would be the form with no hypotheses; in this case the "theorem" form would be the original axiom ax-1 6. We usually show the theorem form without a suffix on its label (e.g. pm2.43 53 versus pm2.43i 49 versus pm2.43d 50). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 20-Mar-2006.)

Theorem2a1d 23 Deduction introducing two antecedents. Two applications of a1d 22. Deduction associated with 2a1 25 and 2a1i 27. (Contributed by BJ, 10-Aug-2020.)

Theorema1i13 24 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorem2a1 25 A double form of ax-1 6. Its associated inference is 2a1i 27. Its associated deduction is 2a1d 23. (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)

Theorema2d 26 Deduction distributing an embedded antecedent. (Contributed by NM, 23-Jun-1994.)

Theorem2a1i 27 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theoremsylcom 28 Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)

Theoremsyl5com 29 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)

Theoremcom12 30 Inference that swaps (commutes) antecedents in an implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

Theoremsyl11 31 A syllogism inference. Commuted form of an instance of syl 14. (Contributed by BJ, 25-Oct-2021.)

Theoremsyl5 32 A syllogism rule of inference. The second premise is used to replace the second antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)

Theoremsyl6 33 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)

Theoremsyl56 34 Combine syl5 32 and syl6 33. (Contributed by NM, 14-Nov-2013.)

Theoremsyl6com 35 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Theoremmpcom 36 Modus ponens inference with commutation of antecedents. (Contributed by NM, 17-Mar-1996.)

Theoremsyli 37 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)

Theoremsyl2im 38 Replace two antecedents. Implication-only version of syl2an 287. (Contributed by Wolf Lammen, 14-May-2013.)

Theoremsyl2imc 39 A commuted version of syl2im 38. Implication-only version of syl2anr 288. (Contributed by BJ, 20-Oct-2021.)

Theorempm2.27 40 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 5-Aug-1993.)

Theoremmpdd 41 A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.)

Theoremmpid 42 A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)

Theoremmpdi 43 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.)

Theoremmpii 44 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)

Theoremsyld 45 Syllogism deduction.

Notice that syld 45 has the same form as syl 14 with added in front of each hypothesis and conclusion. When all theorems referenced in a proof are converted in this way, we can replace with a hypothesis of the proof, allowing the hypothesis to be eliminated with id 19 and become an antecedent. The Deduction Theorem for propositional calculus, e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is always possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Theoremsyldc 46 Syllogism deduction. Commuted form of syld 45. (Contributed by BJ, 25-Oct-2021.)

Theoremmp2d 47 A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theorema1dd 48 Deduction introducing a nested embedded antecedent. (Contributed by NM, 17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.)

Theorempm2.43i 49 Inference absorbing redundant antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)

Theorempm2.43d 50 Deduction absorbing redundant antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)

Theorempm2.43a 51 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.)

Theorempm2.43b 52 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)

Theorempm2.43 53 Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 15-Aug-2004.)

Theoremimim2d 54 Deduction adding nested antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremimim2 55 A closed form of syllogism (see syl 14). Theorem *2.05 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)

Theoremembantd 56 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)

Theorem3syld 57 Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theoremsylsyld 58 A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

Theoremimim12i 59 Inference joining two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 29-Oct-2011.)

Theoremimim1i 60 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

Theoremimim3i 61 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)

Theoremsylc 62 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)

Theoremsyl3c 63 A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)

Theoremsyl6mpi 64 syl6 33 combined with mpi 15. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)

Theoremmpsyl 65 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

Theoremsyl6c 66 Inference combining syl6 33 with contraction. (Contributed by Alan Sare, 2-May-2011.)

Theoremsyldd 67 Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

Theoremsyl5d 68 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.)

Theoremsyl7 69 A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Theoremsyl6d 70 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.) (Revised by NM, 3-Feb-2006.)

Theoremsyl8 71 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Theoremsyl9 72 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

Theoremsyl9r 73 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremimim12d 74 Deduction combining antecedents and consequents. (Contributed by NM, 7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.)

Theoremimim1d 75 Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)

Theoremimim1 76 A closed form of syllogism (see syl 14). Theorem *2.06 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)

Theorempm2.83 77 Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theoremcom23 78 Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

Theoremcom3r 79 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

Theoremcom13 80 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom3l 81 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theorempm2.04 82 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)

Theoremcom34 83 Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM, 25-Apr-1994.)

Theoremcom4l 84 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.)

Theoremcom4t 85 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)

Theoremcom4r 86 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

Theoremcom24 87 Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom14 88 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom45 89 Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom35 90 Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom25 91 Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom5l 92 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

Theoremcom15 93 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

Theoremcom52l 94 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom52r 95 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom5r 96 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)

Theoremjarr 97 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 9-May-2013.)

Theorempm2.86i 98 Inference based on pm2.86 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

Theorempm2.86d 99 Deduction based on pm2.86 100. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

Theorempm2.86 100 Converse of axiom ax-2 7. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

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