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PART 1  FIRST ORDER LOGIC WITH EQUALITY

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.

Theorema1ii 1 (Note: This inference rule and the next one, idi 2, will normally never appear in a completed proof. It can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.)

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 program's Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user to 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 '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 'let' command to pre-assign the conclusion of the independent subproof to a1ii.2. (3) After the independent subproof is complete, use '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 'minimize *' to clean up (discard) all a1ii 1 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. (Contributed by NM, 7-Feb-2006.)

Theoremidi 2 Inference form of id 19. This inference rule, which requires no axioms for its proof, 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 and can be removed using the 'minimize *' command in the metamath program's Proof Assistant. (Contributed by Alan Sare, 31-Dec-2011.)

1.2  Propositional calculus

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 1392 and wel 1394).

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-1 5 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 6 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 pm5.41 240. (Contributed by NM, 5-Aug-1993.)

Axiomax-mp 7 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.

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, 5-Aug-1993.)

1.2.3  Logical implication

The results in this section are based on implication only, and only use ax-1 5, ax-2 6, and ax-mp 7. 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 5. See a1d 22 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 40. (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 6. (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 49 and imim1 70, 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 97) 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 5. We usually show the theorem form without a suffix on its label (e.g. pm2.43 47 vs. pm2.43i 43 vs. pm2.43d 44). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 20-Mar-2006.)

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

Theorem2a1i 24 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theoremsylcom 25 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 26 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)

Theoremcom12 27 Inference that swaps (commutes) antecedents in an implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

Theoremsyl5 28 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 29 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 30 Combine syl5 28 and syl6 29. (Contributed by NM, 14-Nov-2013.)

Theoremsyl6com 31 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Theoremmpcom 32 Modus ponens inference with commutation of antecedents. (Contributed by NM, 17-Mar-1996.)

Theoremsyli 33 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)

Theoremsyl2im 34 Replace two antecedents. Implication-only version of syl2an 273. (Contributed by Wolf Lammen, 14-May-2013.)

Theorempm2.27 35 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 7. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 5-Aug-1993.)

Theoremmpdd 36 A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.)

Theoremmpid 37 A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)

Theoremmpdi 38 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.)

Theoremmpii 39 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)

Theoremsyld 40 Syllogism deduction.

Notice that syld 40 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.)

Theoremmp2d 41 A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theorema1dd 42 Deduction introducing a nested embedded antecedent. (Contributed by NM, 17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.)

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

Theorempm2.43d 44 Deduction absorbing redundant antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)

Theorempm2.43a 45 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.)

Theorempm2.43b 46 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)

Theorempm2.43 47 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 48 Deduction adding nested antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremimim2 49 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 50 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)

Theorem3syld 51 Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theoremsylsyld 52 Virtual deduction rule. (Contributed by Alan Sare, 20-Apr-2011.)

Theoremimim12i 53 Inference joining two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 29-Oct-2011.)

Theoremimim1i 54 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 55 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)

Theoremsylc 56 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)

Theoremsyl3c 57 A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)

Theoremsyl6mpi 58 syl6 29 combined with mpi 15. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)

Theoremmpsyl 59 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

Theoremsyl6c 60 Inference combining syl6 29 with contraction. (Contributed by Alan Sare, 2-May-2011.)

Theoremsyldd 61 Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

Theoremsyl5d 62 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 63 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 64 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 65 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 66 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

Theoremsyl9r 67 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremimim12d 68 Deduction combining antecedents and consequents. (Contributed by NM, 7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.)

Theoremimim1d 69 Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)

Theoremimim1 70 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 71 Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theoremcom23 72 Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

Theoremcom3r 73 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

Theoremcom13 74 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom3l 75 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theorempm2.04 76 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)

Theoremcom34 77 Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM, 25-Apr-1994.)

Theoremcom4l 78 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.)

Theoremcom4t 79 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)

Theoremcom4r 80 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

Theoremcom24 81 Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom14 82 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Theoremcom45 83 Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom35 84 Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom25 85 Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom5l 86 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

Theoremcom15 87 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

Theoremcom52l 88 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom52r 89 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

Theoremcom5r 90 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)

Theoremjarr 91 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 9-May-2013.)

Theorempm2.86i 92 Inference based on pm2.86 94. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

Theorempm2.86d 93 Deduction based on pm2.86 94. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

Theorempm2.86 94 Converse of axiom ax-2 6. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)

Theoremloolin 95 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.)

Theoremloowoz 96 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.)

1.2.4  Logical conjunction and logical equivalence

Syntaxwa 97 Extend wff definition to include conjunction ('and').

Syntaxwb 98 Extend our wff definition to include the biconditional connective.

Axiomax-ia1 99 Left 'and' elimination. One of the axioms of propositional logic. Use its alias simpl 102 instead for naming consistency with set.mm. (New usage is discouraged.) (Contributed by Mario Carneiro, 31-Jan-2015.)

Axiomax-ia2 100 Right 'and' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias simpr 103 instead for naming consistency with set.mm. (New usage is discouraged.)

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