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pmproofs.txt
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pmproofs.txt
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Shortest known proofs of the propositional calculus theorems from
-----------------------------------------------------------------
_Principia Mathematica_
-----------------------
------------------------------------------------------------------------
PUBLIC DOMAIN
This file was released into the Public Domain per the Creative Commons
Public Domain Dedication by Norman Megill and Gregory Bush, on Feb. 16,
2007. See http://creativecommons.org/licenses/publicdomain/
The public domain release applies worldwide. In case this is not
legally possible, the right is granted to use the work for any purpose,
without any conditions, unless such conditions are required by law.
------------------------------------------------------------------------
Revision history
13-Feb-2013: Shorter proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76,
*4.83, *5.33, *5.35, *5.36, and meredith were found by Scott Fenton
(SF).
23-Jan-2012: Added a clarification.
30-Aug-2011: Shorter proofs of *3.33, *3.45, *4.36, and meredith
were found by Scott Fenton (SF).
12-Jun-2010: A proof of Meredith's single axiom for propositional
calculus was added. Contributed by Scott Fenton.
6-Mar-2008: Corrected theorems *2.36, *2.37, and *2.38, which were
shown with v expanded into ~ and ->. (The "Result of proof" and the
proof itself were still correct, however).
4-May-2004: Gregory Bush found shorter proofs for 67 of the original
193 proofs. This file has been updated with them. When you see a
comment like "(GB 35->25)" after the theorem, it means the original
proof had 35 steps and Greg found a shorter version with only 25 steps.
------------------------------------------------------------------------
This file contains proofs of all 193 theorems of propositional calculus
in Whitehead and Russell's _Principia Mathematica_, directly from axioms
ax-1, ax-2, ax-3, and ax-mp in the Metamath Solitaire applet
(http://metamath.org). These are the shortest known direct proofs. If
you find a shorter one, let me know at nm (at) alum (dot) mit (dot) edu
(include the word "Metamath" in the subject).
In these proofs, the resulting theorem is in a form which has all
abbreviations expanded, so instead of "(~ P v P)" for theorem *2.1 you
will see "(~ ~ P -> P)" or in Metamath Solitaire's ASCII notation,
"(-. -. P -> P)". To get "(~ P v P)" we would apply the definition of
logical OR, which is shown in the applet using: Select Logic Family ->
Propositional Calculus + Definitions -> Axiom Information -> Axioms ->
df-or. If we wanted we could extend the proof to apply the definition
directly in the proof, but that is somewhat tedious and was not our
purpose here.
In addition, sometimes "more general" theorems are proved. For example,
the proof of theorem *1.6 actually proves
"((P -> Q) -> ((R -> P) -> (R -> Q)))" in the applet. We obtain theorem
*1.6 by substituting "-. R" for "R", then applying the definition of
logical OR.
The notation for proofs is as follows. For theorem *1.6, for example,
the proof shown is "DD2D121". To enter this into the applet, you enter
the steps in _reverse_ order i.e. "121D2DD". The character mapping is
as follows: "1" is ax-1, "2" is ax-2, "3" is ax-3, and "D" is ax-mp.
Thus this proof would be entered into the applet as:
ax-1 ax-2 ax-1 ax-mp ax-2 ax-mp ax-mp
As a historical note, the notation in "DD2D121" is called "condensed
detachment" and expresses the proof in Polish notation. It was invented
by logician C. A. Meredith in the 1950's. For more information on this
notation you can reference the article "A Finitely Axiomatized
Formalization of Predicate Calculus with Equality" linked to from the
Metamath Home Page. The Appendix in that article shows how to write a
very simple program for verifying these proofs. For longer proofs such
a program is much more convenient than repeated clicks on the applet.
If this interests you, I have a hastily-written C program, drule.c,
that implements the algorithm and which was used to verify these proofs.
It is unpolished and probably not suitable for general release, but it
has no known bugs. Download here:
http://us.metamath.org/downloads/drule.c
To compile:
gcc drule.c -o drule
To run:
./drule < pmproofs.txt > newpmproofs.txt
where "pmproofs.txt" is the name of this file. Then drule.c will verify
each proof below and also regenerate the "Result of proof" and the
number of steps in the output file "newpmproofs.txt". Unfortunately, it
will also discard this comment header, which must be put back by hand.
See comments at the top of drule.c for more information. Note:
everything up to and including the LAST LINE OF DASHES below is treated
as a comment, i.e. discarded, by the drule.c proof checker. In
addition, due to the crudeness of drule.c, the text above the last line
of dashes MUST NOT CONTAIN SEMICOLONS. This is not a bug but is an
intentional design shortcut made to write the program quickly.
(Specifically: Until there is a semicolon identifying the end of the
first statement, no processing occurs. When a line of dashes is
encountered, reading of the input restarts.) If you want to enhance the
program, be my guest. :)
------------------------------------------------------------------------
Clarification added 23-Jan-2012
-------------------------------
The general idea behind this page is to present the shortest known
formal proof as it is usually defined in most logic textbooks for a
system with axiom schemes. In such a system, a proof step is either an
axiom (i.e. a specific _instance_ of an axiom scheme) or the result of
an inference rule applied to previous steps.
The D-rule, as used in the literature, results in a (most general)
_scheme_. With the aid of a more complex notation, a proof could be
built up from D-proof pieces that are referenced multiple times as
subproofs, resulting in fewer total steps. However, on this page, the
D-notation is intended instead merely as a convenient way to abbreviate
textbook formal proofs, in which it is rare that a specific _instance_
of an axiom scheme repeats in the formal proof. In the occasional case
where that happens, a slightly shorter formal proof could be
reconstructed from the D-proof by expanding it and looking for repeated
instances, obtaining a formal proof whose length would be slightly less
than the D-proof length. (I haven't looked at how often such repeated
instances occur or even if they occur at all in these proofs.)
To avoid such quibbles, we can restate our goal as simply being the
shortest complete formal proof expressible in pure D-notation.
------------------------------------------------------------------------
File format
-----------
For each theorem below, we list: (1) the theorem as it appears in
_Principia Mathematica_ along with P.M.'s theorem number, (2) the
unabbreviated result of the proof as it will appear in the Metamath
Solitaire Java applet, and (3) the proof in the condensed detachment
notation explained above. A semicolon ends the theorem or proof, and an
exclamation point means the rest of the line is a comment. All
proofs were verified and formatted by drule.c as explained above.
In the condensed detachment notation, "D" stands for the second of its
two arguments detached from the first (i.e. the first argument is the
major premise, and the second the minor premise, of modus ponens). "1",
"2", and "3" stand for the three axioms of the system called P_2
(attributed to Lukasiewicz and popularized by Church):
ax-1 (P -> (Q -> P))
ax-2 ((P -> (Q -> R)) -> ((P -> Q) -> (P -> R)))
ax-3 ((~ P -> ~ Q) -> (Q -> P))
---------------------------------------------------------------------------
For drule.c: THERE MUST BE NO SEMICOLONS IN THE ENTIRE BODY OF TEXT ABOVE.
For drule.c: THIS COMMENT HEADER ENDS WITH THE LINE OF DASHES BELOW.
---------------------------------------------------------------------------
((P v P) -> P); ! *1.2 Taut
((~ P -> P) -> P); ! Result of proof
DD2DD2D13D2DD2D1311; ! 19 steps
(Q -> (P v Q)); ! *1.3 Add
(P -> (Q -> P)); ! Result of proof
1; ! 1 step
((P v Q) -> (Q v P)); ! *1.4
((~ P -> Q) -> (~ Q -> P)); ! Result of proof
DD2D13D2D1D3DD2DD2D13DD2D1311; ! 29 steps
((P v (Q v R)) -> (Q v (P v R))); ! *1.5 Assoc
((P -> (Q -> R)) -> (Q -> (P -> R))); ! Result of proof
DD2D1DD22D11DD2D112; ! 19 steps
((Q -> R) -> ((P v Q) -> (P v R))); ! *1.6 Sum
((P -> Q) -> ((R -> P) -> (R -> Q))); ! Result of proof
DD2D121; ! 7 steps
((P -> ~ P) -> ~ P); ! *2.01
((P -> ~ P) -> ~ P); ! Result of proof
DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311; ! 49 steps
(Q -> (P -> Q)); ! *2.02 Simp
(P -> (Q -> P)); ! Result of proof
1; ! 1 step
((P -> ~ Q) -> (Q -> ~ P)); ! *2.03
((P -> ~ Q) -> (Q -> ~ P)); ! Result of proof
DD2D13DD2D1DD22D2DD2D13DD2D1311; ! 31 steps
((P -> (Q -> R)) -> (Q -> (P -> R))); ! *2.04 Comm
((P -> (Q -> R)) -> (Q -> (P -> R))); ! Result of proof
DD2D1DD22D11DD2D112; ! 19 steps
((Q -> R) -> ((P -> Q) -> (P -> R))); ! *2.05 Syll
((P -> Q) -> ((R -> P) -> (R -> Q))); ! Result of proof
DD2D121; ! 7 steps
((P -> Q) -> ((Q -> R) -> (P -> R))); ! *2.06 Syll
((P -> Q) -> ((Q -> R) -> (P -> R))); ! Result of proof
DD2D1D2DD2D1211; ! 15 steps
(P -> (P v P)); ! *2.07
(P -> (Q -> P)); ! Result of proof
1; ! 1 step
(P -> P); ! *2.08 Id
(P -> P); ! Result of proof
DD211; ! 5 steps
(~ P v P); ! *2.1
(~ ~ P -> P); ! Result of proof
DD2DD2D13DD2D1311; ! 17 steps
(P v ~ P); ! *2.11
(P -> P); ! Result of proof
DD211; ! 5 steps
(P -> ~ ~ P); ! *2.12
(P -> ~ ~ P); ! Result of proof
D3DD2DD2D13DD2D1311; ! 19 steps
(P v ~ ~ ~ P); ! *2.13
(P -> ~ ~ P); ! Result of proof
D3DD2DD2D13DD2D1311; ! 19 steps
(~ ~ P -> P); ! *2.14
(~ ~ P -> P); ! Result of proof
DD2DD2D13DD2D1311; ! 17 steps
((~ P -> Q) -> (~ Q -> P)); ! *2.15 Transp
((~ P -> Q) -> (~ Q -> P)); ! Result of proof
DD2D13D2D1D3DD2DD2D13DD2D1311; ! 29 steps
((P -> Q) -> (~ Q -> ~ P)); ! *2.16
((P -> Q) -> (~ Q -> ~ P)); ! Result of proof
DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311
; ! 59 steps
((~ Q -> ~ P) -> (P -> Q)); ! *2.17 Transp
((~ P -> ~ Q) -> (Q -> P)); ! Result of proof
3; ! 1 step
((~ P -> P) -> P); ! *2.18
((~ P -> P) -> P); ! Result of proof
DD2DD2D13D2DD2D1311; ! 19 steps
(P -> (P v Q)); ! *2.2
(P -> (~ P -> Q)); ! Result of proof
DD2D1D2DD2D1311; ! 15 steps
(~ P -> (P -> Q)); ! *2.21
(~ P -> (P -> Q)); ! Result of proof
DD2D131; ! 7 steps
(P -> (~ P -> Q)); ! *2.24
(P -> (~ P -> Q)); ! Result of proof
DD2D1D2DD2D1311; ! 15 steps
(P v ((P v Q) -> Q)); ! *2.25
(P -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD2111; ! 13 steps
(~ P v ((P -> Q) -> Q)); ! *2.26 (GB 35->25)
(~ ~ P -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD211DD2D13DD2D131; ! 25 steps
(P -> ((P -> Q) -> Q)); ! *2.27
(P -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD2111; ! 13 steps
((P v (Q v R)) -> (P v (R v Q))); ! *2.3
((P -> (~ Q -> R)) -> (P -> (~ R -> Q))); ! Result of proof
D2D1DD2D13D2D1D3DD2DD2D13DD2D1311; ! 33 steps
((P v (Q v R)) -> ((P v Q) v R)); ! *2.31
((P -> (~ Q -> R)) -> (~ (P -> Q) -> R)); ! Result of proof
DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D13D
2D1D3DD2DD2D13DD2D1311; ! 91 steps
(((P v Q) v R) -> (P v (Q v R))); ! *2.32
((~ (P -> Q) -> R) -> (P -> (~ Q -> R))); ! Result of proof
DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D13D
2D1D3DD2DD2D13DD2D1311; ! 91 steps
((Q -> R) -> ((P v Q) -> (R v P))); ! *2.36
((P -> Q) -> ((~ R -> P) -> (~ Q -> R))); ! Result of proof
DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121; ! 45 steps
((Q -> R) -> ((Q v P) -> (P v R))); ! *2.37
((P -> Q) -> ((~ P -> R) -> (~ R -> Q))); ! Result of proof
DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121; ! 53 steps
((Q -> R) -> ((Q v P) -> (R v P))); ! *2.38
((P -> Q) -> ((~ P -> R) -> (~ Q -> R))); ! Result of proof
DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D
13DD2D1311; ! 79 steps
((P v (P v Q)) -> (P v Q)); ! *2.4
((P -> (P -> Q)) -> (P -> Q)); ! Result of proof
DD22D21; ! 7 steps
((Q v (P v Q)) -> (P v Q)); ! *2.41
((~ P -> (Q -> P)) -> (Q -> P)); ! Result of proof
DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD22D11DD2D112; ! 47 steps
((~ P v (P -> Q)) -> (P -> Q)); ! *2.42
((~ ~ P -> (P -> Q)) -> (P -> Q)); ! Result of proof
DD2D1DD22D21DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 43 steps
((P -> (P -> Q)) -> (P -> Q)); ! *2.43
((P -> (P -> Q)) -> (P -> Q)); ! Result of proof
DD22D21; ! 7 steps
(~ (P v Q) -> ~ P); ! *2.45
(~ (P -> Q) -> P); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 33 steps
(~ (P v Q) -> ~ Q); ! *2.46
(~ (P -> Q) -> ~ Q); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D131; ! 39 steps
(~ (P v Q) -> (~ P v Q)); ! *2.47 (GB 35->31)
(~ (P -> Q) -> (~ P -> R)); ! Result of proof
DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps
(~ (P v Q) -> (P v ~ Q)); ! *2.48 (GB 45->43)
(~ (P -> Q) -> (R -> ~ Q)); ! Result of proof
DD2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D131; ! 43 steps
(~ (P v Q) -> (~ P v ~ Q)); ! *2.49 (GB 35->31)
(~ (P -> Q) -> (~ P -> R)); ! Result of proof
DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps
(~ (P -> Q) -> (~ P -> Q)); ! *2.5 (GB 35->31)
(~ (P -> Q) -> (~ P -> R)); ! Result of proof
DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps
(~ (P -> Q) -> (P -> ~ Q)); ! *2.51 (GB 45->43)
(~ (P -> Q) -> (R -> ~ Q)); ! Result of proof
DD2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D131; ! 43 steps
(~ (P -> Q) -> (~ P -> ~ Q)); ! *2.52 (GB 35->31)
(~ (P -> Q) -> (~ P -> R)); ! Result of proof
DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps
(~ (P -> Q) -> (Q -> P)); ! *2.521 (GB 29->25)
(~ (P -> Q) -> (Q -> R)); ! Result of proof
DD2D1DD22D11DD2D11DD2D131; ! 25 steps
((P v Q) -> (~ P -> Q)); ! *2.53
(P -> P); ! Result of proof
DD211; ! 5 steps
((~ P -> Q) -> (P v Q)); ! *2.54
(P -> P); ! Result of proof
DD211; ! 5 steps
(~ P -> ((P v Q) -> Q)); ! *2.55
(P -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD2111; ! 13 steps
(~ Q -> ((P v Q) -> P)); ! *2.56 (GB 37->35)
(~ P -> ((~ Q -> P) -> Q)); ! Result of proof
DD2D1D2DD2D1D231DD2D12DD2D11DD2D131; ! 35 steps
((~ P -> Q) -> ((P -> Q) -> Q)); ! *2.6
((~ P -> Q) -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD
2D13DD2D1311DD2D1211; ! 89 steps
((P -> Q) -> ((~ P -> Q) -> Q)); ! *2.61
((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof
DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2
D1311DD2D121; ! 81 steps
((P v Q) -> ((P -> Q) -> Q)); ! *2.62
((~ P -> Q) -> ((P -> Q) -> Q)); ! Result of proof
DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD
2D13DD2D1311DD2D1211; ! 89 steps
((P -> Q) -> ((P v Q) -> Q)); ! *2.621
((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof
DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2
D1311DD2D121; ! 81 steps
((P v Q) -> ((~ P v Q) -> Q)); ! *2.63
((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof
DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2
D1311DD2D121; ! 81 steps
((P v Q) -> ((P v ~ Q) -> P)); ! *2.64 (GB 61->39)
((~ P -> Q) -> ((~ P -> ~ Q) -> P)); ! Result of proof
DD2D13D2DD2D13DD2D1D2DD2DD2D13DD2D13111; ! 39 steps
((P -> Q) -> ((P -> ~ Q) -> ~ P)); ! *2.65
((P -> Q) -> ((P -> ~ Q) -> ~ P)); ! Result of proof
DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2DD2D13DD2D1D2DD2DD2D13DD2D13111
; ! 69 steps
(((P v Q) -> Q) -> (P -> Q)); ! *2.67
(((~ P -> Q) -> R) -> (P -> R)); ! Result of proof
DD2DD2D121D1DD2D1D2DD2D1311; ! 27 steps
(((P -> Q) -> Q) -> (P v Q)); ! *2.68
(((P -> Q) -> R) -> (~ P -> R)); ! Result of proof
DD2DD2D121D1DD2D131; ! 19 steps
(((P -> Q) -> Q) -> ((Q -> P) -> P)); ! *2.69
(((P -> Q) -> R) -> ((R -> P) -> P)); ! Result of proof
DD2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1D2DD2D1211DD2DD2D121D1DD2D131
; ! 67 steps
((P -> Q) -> (((P v Q) v R) -> (Q v R))); ! *2.73
((P -> Q) -> ((~ (~ P -> Q) -> R) -> (~ Q -> R))); ! Result of proof
DD2D1DD2D1D2DD2D1211DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD
2DD2D13DD2D1311DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D
1D3DD2DD2D13DD2D1311DD2D121; ! 165 steps
((Q -> P) -> (((P v Q) v R) -> (P v R))); ! *2.74
((P -> Q) -> ((~ (~ Q -> P) -> R) -> (~ Q -> R))); ! Result of proof
DD2D1DD2D1D2DD2D1211DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD
2DD2D13DD2D1311DD2D1D2D1DD2DD2D13D2DD2D1311DD2D121; ! 119 steps
((P v Q) -> ((P v (Q -> R)) -> (P v R))); ! *2.75
((P -> Q) -> ((P -> (Q -> R)) -> (P -> R))); ! Result of proof
DD2D1D221; ! 9 steps
((P v (Q -> R)) -> ((P v Q) -> (P v R))); ! *2.76
((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! Result of proof
2; ! 1 step
((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! *2.77
((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! Result of proof
2; ! 1 step
((Q v R) -> ((~ R v S) -> (Q v S))); ! *2.8 (GB 63->43)
((P -> Q) -> ((~ ~ Q -> R) -> (P -> R))); ! Result of proof
DD2D1DD2D1D2DD2D1211D2D1D3DD2DD2D13DD2D1311; ! 43 steps
((Q -> (R -> S)) -> ((P v Q) -> ((P v R) -> (P v S)))); ! *2.81
((P -> (Q -> R)) -> ((S -> P) -> ((S -> Q) -> (S -> R))))
; ! Result of proof
DD2D1D2D12DD2D121; ! 17 steps
(((P v Q) v R) -> (((P v ~ R) v S) -> ((P v Q) v S))); ! *2.82
((~ (P -> Q) -> R) -> ((~ (P -> ~ R) -> S) -> (~ (P -> Q) -> S)))
; ! Result of proof
DD2D1DD2D1D2DD2D1211D2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD
2DD2D13DD2D1311DD2D131; ! 91 steps
((P -> (Q -> R)) -> ((P -> (R -> S)) -> (P -> (Q -> S)))); ! *2.83
((P -> (Q -> R)) -> ((P -> (R -> S)) -> (P -> (Q -> S))))
; ! Result of proof
DD2D12D2D1DD2D1D2DD2D1211; ! 25 steps
(((P v Q) -> (P v R)) -> (P v (Q -> R))); ! *2.85 (GB 41->37)
(((P -> Q) -> (R -> S)) -> (R -> (Q -> S))); ! Result of proof
DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11; ! 37 steps
(((P -> Q) -> (P -> R)) -> (P -> (Q -> R))); ! *2.86 (GB 41->37)
(((P -> Q) -> (R -> S)) -> (R -> (Q -> S))); ! Result of proof
DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11; ! 37 steps
((P ^ Q) -> ~ (~ P v ~ Q)); ! *3.1 (GB 79->73) (SF 73->69)
(~ (P -> Q) -> ~ (~ ~ P -> Q)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2DD2D12DD2D13DD2D131D1D3DD2DD2D13DD2D1311
; ! 69 steps
(~ (~ P v ~ Q) -> (P ^ Q)); ! *3.11 (GB 73->63)
(~ (~ ~ P -> Q) -> ~ (P -> Q)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131
; ! 63 steps
((~ P v ~ Q) v (P ^ Q)); ! *3.12 (GB 73->63)
(~ (~ ~ P -> Q) -> ~ (P -> Q)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131
; ! 63 steps
(~ (P ^ Q) -> (~ P v ~ Q)); ! *3.13 (GB 47->37)
(~ ~ (P -> Q) -> (~ ~ P -> Q)); ! Result of proof
DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131; ! 37 steps
((~ P v ~ Q) -> ~ (P ^ Q)); ! *3.14
((~ ~ P -> Q) -> ~ ~ (P -> Q)); ! Result of proof
DD2D1D3DD2DD2D13DD2D1311DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 55 steps
(P -> (Q -> (P ^ Q))); ! *3.2
(P -> (Q -> ~ (P -> ~ Q))); ! Result of proof
DD2D13DD2D1D2DD2DD2D13DD2D13111; ! 31 steps
(Q -> (P -> (P ^ Q))); ! *3.21
(P -> (Q -> ~ (Q -> ~ P))); ! Result of proof
DD2D1D2DD2D13DD2D1D2DD2DD2D13DD2D131111; ! 39 steps
((P ^ Q) -> (Q ^ P)); ! *3.22 (GB 79->69)
(~ (P -> ~ Q) -> ~ (Q -> ~ P)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131
; ! 69 steps
~ (P ^ ~ P); ! *3.24
~ ~ (P -> ~ ~ P); ! Result of proof
DD3DD2DD2D13DD2D1311D3DD2DD2D13DD2D1311; ! 39 steps
((P ^ Q) -> P); ! *3.26 Simp
(~ (P -> Q) -> P); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 33 steps
((P ^ Q) -> Q); ! *3.27 Simp
(~ (P -> ~ Q) -> Q); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D13111; ! 27 steps
(((P ^ Q) -> R) -> (P -> (Q -> R))); ! *3.3 Exp (GB 63->59)
((~ (P -> ~ Q) -> R) -> (P -> (Q -> R))); ! Result of proof
DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111
; ! 59 steps
((P -> (Q -> R)) -> ((P ^ Q) -> R)); ! *3.31 Imp
((P -> (Q -> R)) -> (~ (P -> ~ Q) -> R)); ! Result of proof
DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD
2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 121 steps
(((P -> Q) ^ (Q -> R)) -> (P -> R))
; ! *3.33 Syll (GB 113->105) (SF 105->95)
(~ ((P -> Q) -> ~ (Q -> R)) -> (P -> R)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD
2D1D2DD2D12DD2D13DD2D13111; ! 95 steps
(((Q -> R) ^ (P -> Q)) -> (P -> R)); ! *3.34 Syll
(~ ((P -> Q) -> ~ (R -> P)) -> (R -> Q)); ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2
D1311D2D1D3DD2DD2D13DD2D1311DD2D1211; ! 105 steps
((P ^ (P -> Q)) -> Q); ! *3.35 Ass (GB 93->63)
(~ (P -> ~ (P -> Q)) -> Q); ! Result of proof
DD2D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131
; ! 63 steps
(((P ^ Q) -> R) -> ((P ^ ~ R) -> ~ Q)); ! *3.37 Transp
((~ (P -> ~ Q) -> R) -> (~ (P -> ~ ~ R) -> ~ Q)); ! Result of proof
DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1
DD2D1DD2D1DD2D1DD22D11DD2D112D2D1DD2D1D2D1D3DD2DD2D13DD2D13113DD2D1DD
22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311; ! 179 steps
((P ^ Q) -> (P -> Q)); ! *3.4 (GB 33->31)
(~ (P -> ~ Q) -> (R -> Q)); ! Result of proof
DD2D1DD2D13DD22D11DD2D11DD2D131; ! 31 steps
((P -> R) -> ((P ^ Q) -> R)); ! *3.41
((P -> Q) -> (~ (P -> R) -> Q)); ! Result of proof
DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 45 steps
((Q -> R) -> ((P ^ Q) -> R)); ! *3.42
((P -> Q) -> (~ (R -> ~ P) -> Q)); ! Result of proof
DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111; ! 39 steps
(((P -> Q) ^ (P -> R)) -> (P -> (Q ^ R))); ! *3.43 Comp (SF 143->139)
(~ ((P -> Q) -> ~ (P -> R)) -> (P -> ~ (Q -> ~ R)))
; ! Result of proof
D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD
2D1DD2D1DD22D2DD2D13DD2D1311DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111
1; ! 139 steps
(((Q -> P) ^ (R -> P)) -> ((Q v R) -> P)); ! *3.44
(~ ((P -> Q) -> ~ (R -> Q)) -> ((~ P -> R) -> Q)); ! Result of proof
DD2D13DD2DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D
D2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D1
3DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1DD2D1DD2D13DD2D1DD22D2
DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D13111
; ! 271 steps
((P -> Q) -> ((P ^ R) -> (Q ^ R)))
; ! *3.45 Fact (GB 79->71) (SF 71->61)
((P -> Q) -> (~ (P -> R) -> ~ (Q -> R))); ! Result of proof
DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12DD2D13DD2D1311
; ! 61 steps
(((P -> R) ^ (Q -> S)) -> ((P ^ Q) -> (R ^ S))); ! *3.47
(~ ((P -> Q) -> ~ (R -> S)) -> (~ (P -> ~ R) -> ~ (Q -> ~ S)))
; ! Result of proof
DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2DD2D121D1D3D
D2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1D
D2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D13111
; ! 203 steps
(((P -> R) ^ (Q -> S)) -> ((P v Q) -> (R v S)))
; ! *3.48 (GB 245->241)
(~ ((P -> Q) -> ~ (R -> S)) -> ((~ P -> R) -> (~ Q -> S)))
; ! Result of proof
DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D
1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1
DD2DD2D121D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D13
11DD2D1DD2D1D2D1DD2D1D2DD2D1211211; ! 241 steps
((P -> Q) <-> (~ Q -> ~ P)); ! *4.1
~ (((P -> Q) -> (~ Q -> ~ P)) -> ~ ((~ R -> ~ S) -> (S -> R)))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD
2DD2D13DD2D13113; ! 85 steps
((P <-> Q) <-> (~ P <-> ~ Q)); ! *4.11
~ ((~ ((P -> Q) -> ~ (R -> S)) -> ~ ((~ S -> ~ R) -> ~ (~ Q -> ~
P))) -> ~ (~ ((~ T -> ~ U) -> ~ (~ V -> ~ W)) -> ~ ((W -> V) -> ~
(U -> T)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1
DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3D
D2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD
2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2DD2D1DD2D13DD2D1D2DD2
DD2D13DD2D13111DD2D13D3DD2D1D3DD2DD2D13DD2D13111DD2D13D3DD2D1D3DD2DD2
D13DD2D1311DD2D131; ! 363 steps
((P <-> ~ Q) <-> (Q <-> ~ P)); ! *4.12
~ ((~ ((P -> ~ Q) -> ~ (~ R -> S)) -> ~ ((Q -> ~ P) -> ~ (~ S ->
R))) -> ~ (~ ((T -> ~ U) -> ~ (~ V -> W)) -> ~ ((U -> ~ T) -> ~ (~
W -> V)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1
DD2D13DD2D1DD22D2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1
DD2D13D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D1
3DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D3DD2D1
D3DD2DD2D13DD2D1311DD2D131DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311D3DD2D1D3
DD2DD2D13DD2D13111; ! 363 steps
(P <-> ~ ~ P); ! *4.13
~ ((P -> ~ ~ P) -> ~ (~ ~ Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2DD2D13DD2D1311DD2DD2D13DD2D1311
; ! 61 steps
(((P ^ Q) -> R) <-> ((P ^ ~ R) -> ~ Q)); ! *4.14 (GB 325->321)
~ (((~ (P -> ~ Q) -> R) -> (~ (P -> ~ ~ R) -> ~ Q)) -> ~ ((~ (S -> ~
~ T) -> U) -> (~ (S -> U) -> T))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D
1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1DD22D11DD2D112D2D1DD2D1D2D1D3
DD2DD2D13DD2D13113DD2D1DD22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311DD2
D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2DD2D1DD
2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111; ! 321 steps
(((P ^ Q) -> ~ R) <-> ((Q ^ R) -> ~ P)); ! *4.15 (GB 281->277)
~ (((~ (P -> ~ Q) -> R) -> (~ (Q -> R) -> ~ P)) -> ~ ((~ (S -> ~ T)
-> ~ U) -> (~ (U -> ~ S) -> ~ T))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D
1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13
DD2D131111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2
D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D13DD2D1DD22D2DD2D13DD2D1311
3; ! 277 steps
(P <-> P); ! *4.2
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
((P <-> Q) <-> (Q <-> P)); ! *4.21 (GB 183->163)
~ ((~ (P -> ~ Q) -> ~ (Q -> ~ P)) -> ~ (~ (R -> ~ S) -> ~ (S -> ~
R))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D
D2D13DD2D131DD2D13DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D
D2D13DD2D131DD2D13DD2D131; ! 163 steps
(((P <-> Q) ^ (Q <-> R)) -> (P <-> R)); ! *4.22 (GB 329->273)
(~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((Q -> T) -> ~ (U -> R))) -> ~
((P -> T) -> ~ (U -> S))); ! Result of proof
DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D121D3DD2D1D3D3DD2D
1D3DD2DD2D13DD2D13DD2D13DD2D13111DD2D131D3DD2D1DD2D1D3DD2DD2D13DD2D13
11DD2D1D2DD2D1311DD2D131DD2DD2D1DD2D121D3DD2D1DD2D1D3DD2DD2D13DD2D131
1DD2D1D2DD2D13111D3DD2D1D3D3DD2D1D3DD2DD2D13DD2D13DD2D13DD2D131111
; ! 273 steps
(P <-> (P ^ P)); ! *4.24 (GB 91->89)
~ ((P -> ~ (P -> ~ P)) -> ~ (~ (Q -> ~ R) -> R)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1D231DD2D1D2DD2DD2D13DD2D13111D3DD2D1
D3DD2DD2D13DD2D13111; ! 89 steps
(P <-> (P v P)); ! *4.25
~ ((P -> (Q -> P)) -> ~ ((~ R -> R) -> R)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D11DD2DD2D13D2DD2D1311; ! 45 steps
((P ^ Q) <-> (Q ^ P)); ! *4.3 (GB 183->163)
~ ((~ (P -> ~ Q) -> ~ (Q -> ~ P)) -> ~ (~ (R -> ~ S) -> ~ (S -> ~
R))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D
D2D13DD2D131DD2D13DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D
D2D13DD2D131DD2D13DD2D131; ! 163 steps
((P v Q) <-> (Q v P)); ! *4.31
~ (((~ P -> Q) -> (~ Q -> P)) -> ~ ((~ R -> S) -> (~ S -> R)))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D13D2D1D3DD2
DD2D13DD2D1311; ! 83 steps
(((P ^ Q) ^ R) <-> (P ^ (Q ^ R))); ! *4.32 (GB 359->355)
~ ((~ (~ (P -> ~ Q) -> R) -> ~ (P -> ~ ~ (Q -> R))) -> ~ (~ (S -> ~
~ (T -> U)) -> ~ (~ (S -> ~ T) -> U))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2
D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD2D13DD2D
1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D2D1DD2DD2D13DD2D1311DD2D
D2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1D3DD2DD2D13DD2D1
311DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111DD2DD2D
13DD2D1311; ! 355 steps
(((P v Q) v R) <-> (P v (Q v R))); ! *4.33
~ (((~ (P -> Q) -> R) -> (P -> (~ Q -> R))) -> ~ ((S -> (~ T -> U))
-> (~ (S -> T) -> U))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1D
D2D1DD22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D13D2D1D3DD2DD
2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D13D2D1D3DD2DD2D13DD2D1311
; ! 207 steps
((P <-> Q) -> ((P ^ R) <-> (Q ^ R)))
; ! *4.36 (GB 311->291) (SF 291->271)
(~ ((P -> Q) -> ~ (R -> S)) -> ~ ((~ (P -> T) -> ~ (Q -> T)) -> ~ (~
(R -> U) -> ~ (S -> U)))); ! Result of proof
D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2D13DD2D1311DDDD2DD2D12DD2D11DD2D
12DD2D11DD2D121D1DD2D1D2DD2D1211D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2
D13DD2D1311DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12DD2D13DD2D1
311DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12DD2D13DD2D1311
; ! 271 steps
((P <-> Q) -> ((P v R) <-> (Q v R))); ! *4.37 (GB 311->307)
(~ ((P -> Q) -> ~ (R -> S)) -> ~ (((~ P -> T) -> (~ Q -> T)) -> ~
((~ R -> U) -> (~ S -> U)))); ! Result of proof
D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2D13DD2D1311DDDD2DD2D12DD2D11DD2D
12DD2D11DD2D121D1DD2D1D2DD2D1211D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2
D13DD2D1311DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2
D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13
DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 307 steps
(((P <-> R) ^ (Q <-> S)) -> ((P ^ Q) <-> (R ^ S))); ! *4.38
(~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((T -> U) -> ~ (V -> W))) -> ~
((~ (P -> ~ T) -> ~ (Q -> ~ U)) -> ~ (~ (R -> ~ V) -> ~ (S -> ~
W)))); ! Result of proof
DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2D1D
2DD2DD2D13DD2D13111DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D1311DD
2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D1
31DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13
DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2
D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D131
1DD2D131D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131D
D2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2
D13111D3DD2D1D3DD2DD2D13DD2D13111; ! 585 steps
(((P <-> R) ^ (Q <-> S)) -> ((P v Q) <-> (R v S)))
; ! *4.39 (GB 913->901)
(~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((T -> U) -> ~ (V -> W))) -> ~
(((~ P -> T) -> (~ Q -> U)) -> ~ ((~ R -> V) -> (~ S -> W))))
; ! Result of proof
DD2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DDDD2DD2D12DD2D11DD2D12DD2D11DD2D
121D1DD2D1D2DD2D1211D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2DD2D1DD2D13
D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D1DD2D13DD
2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D1D
D2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1DD
2D1D2DD2D1211211DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2
D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2
D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D1DD2D13DD2D1DD22D2DD
2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1DD2D1D2DD2D1211211DD
2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1D2D1D3DD
2DD2D13DD2D13113D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1DD22D11DD2
D112DD2D1DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13
DD2D1311D2D1DD2D1DD22D11DD2D112D2D13DD2D1DD22D11DD2D1123DD2DD2D13DD2D
1311; ! 901 steps
((P ^ (Q v R)) <-> ((P ^ Q) v (P ^ R))); ! *4.4 (SF 359->355)
~ ((~ (P -> ~ (Q -> R)) -> (~ ~ (P -> Q) -> ~ (P -> ~ R))) -> ~ ((~
~ (S -> T) -> ~ (S -> ~ U)) -> ~ (S -> ~ (T -> U))))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D
1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2DD2D13D
D2D1311DD2D1D2DD2DD2D13DD2D13111DD2DD2D13DD2D13111D3DD2D1DD2DD2D1DD2D
13DD2D1D2DD2DD2D13DD2D13111DD2D1D3DD2DD2D13DD2D1311D2D1D3DD2D1D3DD2DD
2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D131DD2DD2D
13DD2D1311; ! 355 steps
((P v (Q ^ R)) <-> ((P v Q) ^ (P v R))); ! *4.41 (SF 275->271)
~ (((P -> ~ (Q -> ~ R)) -> ~ ((P -> Q) -> ~ (P -> R))) -> ~ (~ ((S
-> T) -> ~ (S -> U)) -> (S -> ~ (T -> ~ U)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D
3DD2D1D3DD2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D
1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2
D1DD22D2DD2D13DD2D1311DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D131111
; ! 271 steps
(P <-> ((P ^ Q) v (P ^ ~ Q))); ! *4.42 (GB 175->165)
~ ((P -> (~ ~ (P -> Q) -> ~ (P -> ~ Q))) -> ~ ((~ ~ (R -> S) -> ~ (R
-> T)) -> R)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D1D2D2DD2D1
3DD2D1D2DD2DD2D13DD2D131111D3DD2DD2D1DD2D1D2DD2D13DD2D1D2DD2D13DD2DD2
D13DD2D131111DD2D131DD2D131; ! 165 steps
(P <-> ((P v Q) ^ (P v ~ Q))); ! *4.43
~ ((P -> ~ ((~ P -> Q) -> ~ (~ P -> R))) -> ~ (~ ((~ S -> T) -> ~ (~
S -> ~ T)) -> S)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1
D2DD2D1311DD2D1D2DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D2DD2D13DD2
D1D2DD2DD2D13DD2D131111; ! 161 steps
(P <-> (P v (P ^ Q))); ! *4.44
~ ((P -> (~ P -> Q)) -> ~ ((~ R -> ~ (R -> S)) -> R))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1D2DD2D1311DD2DD2D13D2DD2D1D2DD2D131DD2D
11DD2D1313; ! 79 steps
(P <-> (P ^ (P v Q))); ! *4.45 (GB 161->107)
~ ((P -> ~ (P -> ~ (~ P -> Q))) -> ~ (~ (R -> S) -> R))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2
D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 107 steps
((P ^ Q) <-> ~ (~ P v ~ Q)); ! *4.5 (GB 177->161) (SF 161->157)
~ ((~ (P -> Q) -> ~ (~ ~ P -> Q)) -> ~ (~ (~ ~ R -> S) -> ~ (R ->
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2DD2D12DD2D13DD2
D131D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2D13D
D2D131DD2D13DD2D131; ! 157 steps
(~ (P ^ Q) <-> (~ P v ~ Q)); ! *4.51 (GB 127->117)
~ ((~ ~ (P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> ~ ~ (R ->
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131DD2D1D3
DD2DD2D13DD2D1311DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 117 steps
((P ^ ~ Q) <-> ~ (~ P v Q)); ! *4.52 (GB 231->219)
~ ((~ (P -> ~ ~ Q) -> ~ (~ ~ P -> Q)) -> ~ (~ (~ ~ R -> S) -> ~ (R
-> ~ ~ S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D13DD2D13D
2D1D3DD2DD2D13DD2D13DD2D13DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13
DD2D1311DD2D1DD2D1D2D1DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D1311DD2DD
2D13DD2D1311; ! 219 steps
(~ (P ^ ~ Q) <-> (~ P v Q)); ! *4.53 (GB 181->169)
~ ((~ ~ (P -> ~ ~ Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> ~ ~ (R
-> ~ ~ S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1D2D1DD2DD2D13DD2D1311DD2D1DD22D2DD
2D13DD2D1311DD2DD2D13DD2D1311DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D13D2D1
D3DD2DD2D13DD2D13DD2D13DD2D1311; ! 169 steps
((~ P ^ Q) <-> ~ (P v ~ Q)); ! *4.54
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
(~ (~ P ^ Q) <-> (P v ~ Q)); ! *4.55
~ ((~ ~ P -> P) -> ~ (Q -> ~ ~ Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13DD2D1311D3DD2DD2D13DD2D1311
; ! 61 steps
((~ P ^ ~ Q) <-> ~ (P v Q)); ! *4.56
~ ((~ (P -> ~ ~ Q) -> ~ (P -> Q)) -> ~ (~ (R -> S) -> ~ (R -> ~ ~
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D
13DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D
13DD2D1311DD2DD2D13DD2D1311; ! 165 steps
(~ (~ P ^ ~ Q) <-> (P v Q)); ! *4.57
~ ((~ ~ (P -> ~ ~ Q) -> (P -> Q)) -> ~ ((R -> S) -> ~ ~ (R -> ~ ~
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1D2D1DD2DD2D13DD2D1311DD2DD2D13DD2D1311D
D2D1D3DD2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 115 steps
((P -> Q) <-> (~ P v Q)); ! *4.6
~ (((P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> (R -> S)))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D1311DD2DD2D121D1D3DD2DD
2D13DD2D1311; ! 81 steps
(~ (P -> Q) <-> (P ^ ~ Q)); ! *4.61
~ ((~ (P -> Q) -> ~ (P -> ~ ~ Q)) -> ~ (~ (R -> ~ ~ S) -> ~ (R ->
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D13
DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D
13DD2D1311DD2DD2D13DD2D1311; ! 165 steps
((P -> ~ Q) <-> (~ P v ~ Q)); ! *4.62
~ (((P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> (R -> S)))
; ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D1311DD2DD2D121D1D3DD2DD
2D13DD2D1311; ! 81 steps
(~ (P -> ~ Q) <-> (P ^ Q)); ! *4.63
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
((~ P -> Q) <-> (P v Q)); ! *4.64
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
(~ (~ P -> Q) <-> (~ P ^ ~ Q)); ! *4.65
~ ((~ (P -> Q) -> ~ (P -> ~ ~ Q)) -> ~ (~ (R -> ~ ~ S) -> ~ (R ->
S))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D13
DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D
13DD2D1311DD2DD2D13DD2D1311; ! 165 steps
((~ P -> ~ Q) <-> (P v ~ Q)); ! *4.66
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
(~ (~ P -> ~ Q) <-> (~ P ^ Q)); ! *4.67
~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps
((P -> Q) <-> (P -> (P ^ Q))); ! *4.7
~ (((P -> Q) -> (P -> ~ (P -> ~ Q))) -> ~ ((R -> ~ (S -> ~ T)) -> (R
-> T))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D2DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D3DD2D1
D3DD2DD2D13DD2D13111; ! 89 steps
((P -> Q) <-> (P <-> (P ^ Q))); ! *4.71
~ (((P -> Q) -> ~ ((P -> ~ (P -> ~ Q)) -> ~ (~ (R -> S) -> R))) -> ~
(~ ((T -> ~ (U -> ~ V)) -> W) -> (T -> V))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2DD2D13DD2D1D2DD2DD2D13DD2D13111D1D3D
D2D1D3DD2DD2D13DD2D1311DD2D131D2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D
2D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131
; ! 201 steps
((P -> Q) <-> (Q <-> (P v Q))); ! *4.72
~ (((P -> Q) -> ~ ((R -> (S -> R)) -> ~ ((~ P -> Q) -> Q))) -> ~ (~
(T -> ~ ((~ U -> V) -> W)) -> (U -> W))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1D3DD2DD2DD2D13DD2D1311D11DD2D1D2D1DD2DD
2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121DD
2D1DD2DD2D121D1DD2D1D2DD2D1311D3DD2D1D3DD2DD2D13DD2D13111
; ! 195 steps
(Q -> (P <-> (P ^ Q))); ! *4.73
(P -> ~ ((Q -> ~ (Q -> ~ P)) -> ~ (~ (R -> S) -> R)))
; ! Result of proof
DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2D13DD2D1D2DD2DD2D13D
D2D131111D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 113 steps
(~ P -> (Q <-> (P v Q))); ! *4.74
(P -> ~ ((Q -> (R -> Q)) -> ~ ((P -> S) -> S))); ! Result of proof
DD2D1D3DD2DD2DD2D13DD2D1311D11DD2D1D2DD2111; ! 43 steps
(((P -> Q) ^ (P -> R)) <-> (P -> (Q ^ R))); ! *4.76 (SF 275->271)
~ ((~ ((P -> Q) -> ~ (P -> R)) -> (P -> ~ (Q -> ~ R))) -> ~ ((S -> ~
(T -> ~ U)) -> ~ ((S -> T) -> ~ (S -> U)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13
D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D12D2D1DD2D13
DD2D1D2DD2DD2D13DD2D131111DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1
D3DD2D1D3DD2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111
; ! 271 steps
(((Q -> P) ^ (R -> P)) <-> ((Q v R) -> P)); ! *4.77
~ ((~ ((P -> Q) -> ~ (R -> Q)) -> ((~ P -> R) -> Q)) -> ~ (((~ S ->
T) -> U) -> ~ ((S -> U) -> ~ (T -> U)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D13DD2DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2
DD2D13DD2D1311DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2
D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD
2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1
D3DD2DD2D13DD2D13111DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D121
D1DD2D1D2DD2D1311DD2DD2D121D11; ! 375 steps
(((P -> Q) v (P -> R)) <-> (P -> (Q v R))); ! *4.78 (GB 327->319)
~ (((~ (P -> Q) -> (P -> R)) -> (P -> (~ Q -> R))) -> ~ ((S -> (~ T
-> U)) -> (~ (V -> T) -> (S -> U)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11DD
2D1DD2D1DD22D11DD2D112DD2D1DD2D1DD2D1D2D13DD2D1DD2D1DD22D11DD2D112DD2
D1D2D1D2DD2D1D2D2DD2D13DD2D1D2DD2DD2D13DD2D131111DD2D1DD22D11DD2D112D
2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD22D11DD2D112DD2D1DD2DD2D121D1D
D2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D1312; ! 319 steps
(((Q -> P) v (R -> P)) <-> ((Q ^ R) -> P)); ! *4.79 (GB 547->383)
~ (((~ (P -> Q) -> (R -> Q)) -> (~ (P -> ~ R) -> Q)) -> ~ ((~ (S ->
~ T) -> U) -> (~ (S -> V) -> (T -> U)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1
DD2DD2D121D11DD2D12DD2DD2D121D11DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13
DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1DD2D1D2D1
DD2D1D2D1DD2DD2D13D2DD2D13112D2D1DD2D13DD2D1D2DD2D131D2D11DD2D1DD22D1
1DD2D112DD2D1DD2D1DD2D1DD2DD2D121D1DD2D1DD2D13DD2DD2D121D1DD2D131DD2D
131D2D132DD2D13D2D1D3DD2DD2D13DD2D1311; ! 383 steps
((P -> ~ P) <-> ~ P); ! *4.8
~ (((P -> ~ P) -> ~ P) -> ~ (Q -> (R -> Q))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2
D13111; ! 75 steps
((~ P -> P) <-> P); ! *4.81
~ (((~ P -> P) -> P) -> ~ (Q -> (R -> Q))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13D2DD2D13111; ! 45 steps
(((P -> Q) ^ (P -> ~ Q)) <-> ~ P); ! *4.82 (GB 249->179)
~ ((~ ((P -> Q) -> ~ (P -> ~ Q)) -> ~ P) -> ~ (~ R -> ~ ((R -> S) ->
~ (R -> T)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1D
D2D1DD22D2DD2D13DD2D1311D2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D
13DD2D1D2DD2DD2D13DD2D13111DD2D131DD2D131; ! 179 steps
(((P -> Q) ^ (~ P -> Q)) <-> Q); ! *4.83 (SF 249->245)
~ ((~ ((P -> Q) -> ~ (~ P -> Q)) -> Q) -> ~ (R -> ~ ((S -> R) -> ~
(T -> R)))); ! Result of proof
DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1D