\documentclass{amsart}
    \advance\textheight by 1.75in
    \advance\textwidth by 2in
    \advance\topmargin by -1in
    \advance\oddsidemargin by -1in
    \advance\evensidemargin by -1in
    \usepackage{amssymb}
    \parindent0pt
    \begin{document}
    \textbf{LAT002-0.ax} Lattice theory axioms






$n_1\vee x{=}n_1\qquad\mbox{cnf}(\mbox{join\_1\_and\_x},\mbox{axiom})$

$x\vee x{=}x\qquad\mbox{cnf}(\mbox{join\_x\_and\_x},\mbox{axiom})$

$n_0\vee x{=}x\qquad\mbox{cnf}(\mbox{join\_0\_and\_x},\mbox{axiom})$

$n_0\wedge x{=}n_0\qquad\mbox{cnf}(\mbox{meet\_0\_and\_x},\mbox{axiom})$

$x\wedge x{=}x\qquad\mbox{cnf}(\mbox{meet\_x\_and\_x},\mbox{axiom})$

$n_1\wedge x{=}x\qquad\mbox{cnf}(\mbox{meet\_1\_and\_x},\mbox{axiom})$

$x\wedge y{=}z \ \Rightarrow \ y\wedge x{=}z\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y{=}z \ \Rightarrow \ y\vee x{=}z\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$x\wedge y{=}z \ \Rightarrow \ x\vee z{=}x\qquad\mbox{cnf}(\mbox{absorbtion}_1,\mbox{axiom})$

$x\vee y{=}z \ \Rightarrow \ x\wedge z{=}x\qquad\mbox{cnf}(\mbox{absorbtion}_2,\mbox{axiom})$

$( x\wedge y{=}\mbox{xy}\mbox{ and }y\wedge z{=}\mbox{yz}\mbox{ and }x\wedge \mbox{yz}{=}\mbox{xyz} ) \ \Rightarrow \ \mbox{xy}\wedge z{=}\mbox{xyz}\qquad\mbox{cnf}(\mbox{associativity\_of\_meet}_1,\mbox{axiom})$

$( x\wedge y{=}\mbox{xy}\mbox{ and }y\wedge z{=}\mbox{yz}\mbox{ and }\mbox{xy}\wedge z{=}\mbox{xyz} ) \ \Rightarrow \ x\wedge \mbox{yz}{=}\mbox{xyz}\qquad\mbox{cnf}(\mbox{associativity\_of\_meet}_2,\mbox{axiom})$

$( x\vee y{=}\mbox{xy}\mbox{ and }y\vee z{=}\mbox{yz}\mbox{ and }x\vee \mbox{yz}{=}\mbox{xyz} ) \ \Rightarrow \ \mbox{xy}\vee z{=}\mbox{xyz}\qquad\mbox{cnf}(\mbox{associativity\_of\_join}_1,\mbox{axiom})$

$( x\vee y{=}\mbox{xy}\mbox{ and }y\vee z{=}\mbox{yz}\mbox{ and }\mbox{xy}\vee z{=}\mbox{xyz} ) \ \Rightarrow \ x\vee \mbox{yz}{=}\mbox{xyz}\qquad\mbox{cnf}(\mbox{associativity\_of\_join}_2,\mbox{axiom})$

$( x\wedge z{=}x\mbox{ and }x\vee y{=}x_1\mbox{ and }y\wedge z{=}y_1\mbox{ and }z\wedge x_1{=}z_1 ) \ \Rightarrow \ x\vee y_1{=}z_1\qquad\mbox{cnf}(\mbox{modularity}_1,\mbox{axiom})$

$( x\wedge z{=}x\mbox{ and }x\vee y{=}x_1\mbox{ and }y\wedge z{=}y_1\mbox{ and }x\vee y_1{=}z_1 ) \ \Rightarrow \ z\wedge x_1{=}z_1\qquad\mbox{cnf}(\mbox{modularity}_2,\mbox{axiom})$

$x\wedge y{=}\mbox{meet\_of}(x,y)\qquad\mbox{cnf}(\mbox{meet\_total\_function}_1,\mbox{axiom})$

$( x\wedge y{=}z_1\mbox{ and }x\wedge y{=}z_2 ) \ \Rightarrow \ z_1 = z_2\qquad\mbox{cnf}(\mbox{meet\_total\_function}_2,\mbox{axiom})$

$x\vee y{=}\mbox{join\_of}(x,y)\qquad\mbox{cnf}(\mbox{join\_total\_function}_1,\mbox{axiom})$

$( x\vee y{=}z_1\mbox{ and }x\vee y{=}z_2 ) \ \Rightarrow \ z_1 = z_2\qquad\mbox{cnf}(\mbox{join\_total\_function}_2,\mbox{axiom})$

\medskip\textbf{LAT006-0.ax} Tarski's fixed point theorem (equality) axioms






$\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset\_H},\mbox{v\_order\_H},\mbox{v\_more\_H},\mbox{t\_a},\mbox{t\_z}) \ \Rightarrow \ \mbox{v\_pset} = \mbox{v\_pset\_H}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oext\_\_inject}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset\_H},\mbox{v\_order\_H},\mbox{v\_more\_H},\mbox{t\_a},\mbox{t\_z}) \ \Rightarrow \ \mbox{v\_order} = \mbox{v\_order\_H}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oext\_\_inject}_1,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset\_H},\mbox{v\_order\_H},\mbox{v\_more\_H},\mbox{t\_a},\mbox{t\_z}) \ \Rightarrow \ \mbox{v\_more} = \mbox{v\_more\_H}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oext\_\_inject}_2,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_y},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}),\mbox{t\_a},\mbox{t\_z}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oselect\_\_convs\_\_1}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_y},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}),\mbox{t\_a},\mbox{t\_z}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oselect\_\_convs\_\_2}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Omore}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oselect\_\_convs\_\_3}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset\_\_update}(\mbox{v\_pset\_H},\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}),\mbox{t\_a},\mbox{t\_z}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset\_H},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oupdate\_\_convs\_\_1}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Oorder\_\_update}(\mbox{v\_order\_H},\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}),\mbox{t\_a},\mbox{t\_z}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order\_H},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oupdate\_\_convs\_\_2}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Omore\_\_update}(\mbox{v\_more\_H},\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more},\mbox{t\_a},\mbox{t\_z}),\mbox{t\_z},\mbox{t\_a}) = \mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_pset},\mbox{v\_order},\mbox{v\_more\_H},\mbox{t\_a},\mbox{t\_z})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opotype\_Oupdate\_\_convs\_\_3}_0,\mbox{axiom})$

\medskip\textbf{LAT003-0.ax} Ortholattice theory (equality) axioms






$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$x\vee ( x\wedge y ) = x\qquad\mbox{cnf}(\mbox{absorption}_2,\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( x\wedge y )\wedge z = x\wedge ( y\wedge z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet},\mbox{axiom})$

$( x\vee y )\vee z = x\vee ( y\vee z )\qquad\mbox{cnf}(\mbox{associativity\_of\_join},\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{complement\_involution},\mbox{axiom})$

$x\vee ( y\vee y' ) = y\vee y'\qquad\mbox{cnf}(\mbox{join\_complement},\mbox{axiom})$

$x\wedge y = ( x'\vee y' )'\qquad\mbox{cnf}(\mbox{meet\_complement},\mbox{axiom})$

\medskip\textbf{LAT005-0.ax} Weakly Associative Lattices theory (equality) axioms






$x\wedge x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet},\mbox{axiom})$

$x\vee x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_join},\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( ( x\vee y )\wedge ( z\vee y ) )\wedge y = y\qquad\mbox{cnf}(\mbox{wal}_1,\mbox{axiom})$

$( ( x\wedge y )\vee ( z\wedge y ) )\vee y = y\qquad\mbox{cnf}(\mbox{wal}_2,\mbox{axiom})$

\medskip\textbf{LAT001-2.ax} Lattice theory complement (equality) axioms






$x'{=}y \ \Rightarrow \ x\wedge y = n_0\qquad\mbox{cnf}(\mbox{complement\_meet},\mbox{axiom})$

$x'{=}y \ \Rightarrow \ x\vee y = n_1\qquad\mbox{cnf}(\mbox{complement\_join},\mbox{axiom})$

$( x\wedge y = n_0\mbox{ and }x\vee y = n_1 ) \ \Rightarrow \ x'{=}y\qquad\mbox{cnf}(\mbox{meet\_join\_complement},\mbox{axiom})$

\medskip\textbf{LAT001-4.ax} Lattice theory unique complementation (equality) axioms






$x\vee x' = 1\qquad\mbox{cnf}(\mbox{complement\_join},\mbox{axiom})$

$x\wedge x' = 0\qquad\mbox{cnf}(\mbox{complement\_meet},\mbox{axiom})$

$( x\wedge y = 0\mbox{ and }x\vee y = 1 ) \ \Rightarrow \ x' = y\qquad\mbox{cnf}(\mbox{meet\_join\_complement},\mbox{axiom})$

\medskip\textbf{LAT004-0.ax} Quasilattice theory (equality) axioms






$x\wedge x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet},\mbox{axiom})$

$x\vee x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_join},\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( x\wedge y )\wedge z = x\wedge ( y\wedge z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet},\mbox{axiom})$

$( x\vee y )\vee z = x\vee ( y\vee z )\qquad\mbox{cnf}(\mbox{associativity\_of\_join},\mbox{axiom})$

$( x\wedge ( y\vee z ) )\vee ( x\wedge y ) = x\wedge ( y\vee z )\qquad\mbox{cnf}(\mbox{quasi\_lattice}_1,\mbox{axiom})$

$( x\vee ( y\wedge z ) )\wedge ( x\vee y ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{quasi\_lattice}_2,\mbox{axiom})$

\medskip\textbf{LAT001-0.ax} Lattice theory (equality) axioms






$x\wedge x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet},\mbox{axiom})$

$x\vee x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_join},\mbox{axiom})$

$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption}_1,\mbox{axiom})$

$x\vee ( x\wedge y ) = x\qquad\mbox{cnf}(\mbox{absorption}_2,\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( x\wedge y )\wedge z = x\wedge ( y\wedge z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet},\mbox{axiom})$

$( x\vee y )\vee z = x\vee ( y\vee z )\qquad\mbox{cnf}(\mbox{associativity\_of\_join},\mbox{axiom})$

\medskip\textbf{LAT001-3.ax} Lattice theory unique complement (equality) axioms






$\mbox{unique\_complement}(x,y) \ \Rightarrow \ x'{=}y\qquad\mbox{cnf}(\mbox{unique\_complement}_1,\mbox{axiom})$

$( \mbox{unique\_complement}(x,y)\mbox{ and }x'{=}z ) \ \Rightarrow \ z = y\qquad\mbox{cnf}(\mbox{unique\_complement}_2,\mbox{axiom})$

$x'{=}y \ \Rightarrow \ ( \mbox{unique\_complement}(x,y)\mbox{ or }x'{=}f(x,y) )\qquad\mbox{cnf}(\mbox{unique\_complement}_3,\mbox{axiom})$

$( x'{=}y\mbox{ and }f(x,y) = y ) \ \Rightarrow \ \mbox{unique\_complement}(x,y)\qquad\mbox{cnf}(\mbox{unique\_complement}_4,\mbox{axiom})$

\medskip\textbf{LAT001-1.ax} Lattice theory modularity (equality) axioms






$x\wedge n_0 = n_0\qquad\mbox{cnf}(\mbox{x\_meet}_0,\mbox{axiom})$

$x\vee n_0 = x\qquad\mbox{cnf}(\mbox{x\_join}_0,\mbox{axiom})$

$x\wedge n_1 = x\qquad\mbox{cnf}(\mbox{x\_meet}_1,\mbox{axiom})$

$x\vee n_1 = n_1\qquad\mbox{cnf}(\mbox{x\_join}_1,\mbox{axiom})$

$x\wedge z = x \ \Rightarrow \ z\wedge ( x\vee y ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{modular},\mbox{axiom})$

\medskip\textbf{LAT006-1.ax} Tarski's fixed point theorem GLB (equality) axioms






$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_Ix1\_M\_Ay1\_J\_A\_58\_Aorder\_A\_Idual\_Acl\_J\_A\_61\_61\_A\_Iy1\_M\_Ax1\_J\_A\_58\_Aorder\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_Ix1\_M\_Ay1\_J\_A\_58\_Aorder\_A\_Idual\_Acl\_J\_A\_61\_61\_A\_Iy1\_M\_Ax1\_J\_A\_58\_Aorder\_Acl}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT246-1.p} Equation H60 is Huntington by implication

Show that H60 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee z )\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{60},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT061-1.p} Weakly Associative Lattices theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT005-0.ax'})$

\medskip\textbf{LAT196-1.p} Equation H39 is Huntington by distributivity

Show that H39 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{39},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT192-1.p} Equation H22 is Huntington by distributivity

Show that H22 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( z\vee ( x\wedge y ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{22},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT025-1.p} Non-uniqueness of meet (dually join) in TNL

Let's say we have a ternary near-lattice (TNL) with two meet
 operations, say meet1 and meet2.  In other words, {join,meet1}
 and {join,meet2} are TNLs.  Are the two meets necessarily
 the same?  No, they aren't.  Here is a counterexample.




$x\wedge x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet},\mbox{axiom})$

$x\vee x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_join},\mbox{axiom})$

$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption}_1,\mbox{axiom})$

$x\vee ( x\wedge y ) = x\qquad\mbox{cnf}(\mbox{absorption}_2,\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$x\vee ( y\wedge ( x\wedge z ) ) = x\qquad\mbox{cnf}(\mbox{tnl}_1,\mbox{axiom})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{tnl}_2,\mbox{axiom})$

$x\wedge_2 x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet}_2,\mbox{axiom})$

$x\wedge_2 ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption1}_2,\mbox{axiom})$

$x\vee ( x\wedge_2 y ) = x\qquad\mbox{cnf}(\mbox{absorption2}_2,\mbox{axiom})$

$x\wedge_2 y = y\wedge_2 x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet}_2,\mbox{axiom})$

$x\vee ( y\wedge_2 ( x\wedge_2 z ) ) = x\qquad\mbox{cnf}(\mbox{tnl\_1}_2,\mbox{axiom})$

$x\wedge_2 ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{tnl\_2}_2,\mbox{axiom})$

$a\wedge b\ne a\wedge_2 b\qquad\mbox{cnf}(\mbox{prove\_meets\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{LAT212-1.p} Equation H70 is Huntington by distributivity

Show that H70 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( y\wedge ( x\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{70},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT215-1.p} Equation H80 is Huntington by distributivity

Show that H80 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee ( y\wedge ( x\vee z ) ) ) ) ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{equation\_H}_{80},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT247-1.p} Equation H61 is Huntington by implication

Show that H61 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( x\vee y )\wedge ( ( x\wedge y )\vee z ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{61},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT187-1.p} Equation H15 is Huntington by distributivity

Show that H15 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge y )\vee ( ( x\wedge z )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{15},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT095-1.p} Axiom for weakly associative lattices (WAL), part 4






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$b\vee a\ne a\vee b\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT115-1.p} Huntington equation H59 is independent of H55

Show that Huntington equation H55 does not imply Huntington
 equation H59 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$a\wedge ( ( b\vee c )\wedge ( b\vee d ) )\ne a\wedge ( b\vee ( ( b\vee d )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{59},\mbox{negated\_conjecture})$

\medskip\textbf{LAT024-1.p} Meet (dually join) is not necessarily unique for quasilattices.

Let's say we have a quasilattice with two meet operations, say
 meet1 and meet2.  In other words, {join,meet1} is a lattice,
 and {join,meet2} is a lattice.  Then, we can show that the
 two meet operations not necessarily the same.




$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$x\wedge_2 x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet}_2,\mbox{axiom})$

$x\wedge_2 y = y\wedge_2 x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet}_2,\mbox{axiom})$

$( x\wedge_2 y )\wedge_2 z = x\wedge_2 ( y\wedge_2 z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet}_2,\mbox{axiom})$

$( x\wedge_2 ( y\vee z ) )\vee ( x\wedge_2 y ) = x\wedge_2 ( y\vee z )\qquad\mbox{cnf}(\mbox{quasi\_lattice1}_2,\mbox{axiom})$

$( x\vee ( y\wedge_2 z ) )\wedge_2 ( x\vee y ) = x\vee ( y\wedge_2 z )\qquad\mbox{cnf}(\mbox{quasi\_lattice2}_2,\mbox{axiom})$

$a\wedge b\ne a\wedge_2 b\qquad\mbox{cnf}(\mbox{prove\_meets\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{LAT225-1.p} Equation H10 is Huntington by implication

Show that H10 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{10},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT240-1.p} Equation H50 is Huntington by implication

Show that H50 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT238-1.p} Equation H42 is Huntington by implication

Show that H42 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\vee ( x\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{42},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT102-1.p} Huntington equation H4 is independent of H7

Show that Huntington equation H7 does not imply Huntington
 equation H4 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_7,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\vee d ) ) )\ne a\wedge ( b\vee ( ( a\vee ( b\wedge d ) )\wedge ( c\vee d ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT189-1.p} Equation H17 is Huntington by distributivity

Show that H17 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{17},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT230-1.p} Equation H18 is Huntington by implication

Show that H18 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( x\wedge y )\vee ( ( x\wedge z )\vee ( y\wedge ( x\vee z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{18},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT154-1.p} Huntington equation H42 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\vee ( x\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{42},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT190-1.p} Equation H18 is Huntington by distributivity

Show that H18 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( x\wedge y )\vee ( ( x\wedge z )\vee ( y\wedge ( x\vee z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{18},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT137-1.p} Huntington equation H40\_dual is independent of H69

Show that Huntington equation H69 does not imply Huntington
 equation H40\_dual in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge ( z\vee ( x\wedge y ) ) )\vee ( x\wedge ( y\vee ( x\wedge z ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{69},\mbox{axiom})$

$a\vee ( b\wedge ( c\vee ( a\wedge d ) ) )\ne a\vee ( b\wedge ( c\vee ( d\wedge ( c\vee ( a\wedge b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H40\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT062-1.p} E51 does not necessarily hold in ortholattices






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a'\vee a = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a'\wedge a = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$a\wedge b = ( a'\vee b' )'\qquad\mbox{cnf}(\mbox{compatibility},\mbox{axiom})$

$( a\vee b' )\wedge ( ( ( a\wedge b )\vee ( a'\wedge b ) )\vee ( a'\wedge b' ) )\ne ( a\wedge b )\vee ( a'\wedge b' )\qquad\mbox{cnf}(\mbox{prove\_e}_{51},\mbox{negated\_conjecture})$

\medskip\textbf{LAT241-1.p} Equation H51 is Huntington by implication

Show that H51 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge u ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{51},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT281-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Otrans\_\_def}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT150-1.p} Huntington equation H39 implies H40






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{39},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LAT020-1.p} Self-dual distributivity for quasilattices.






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$( ( ( x\wedge y )\vee z )\wedge y )\vee ( z\wedge x ) = ( ( ( x\vee y )\wedge z )\vee y )\wedge ( z\vee x )\qquad\mbox{cnf}(\mbox{self\_dual\_distributivity},\mbox{hypothesis})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT007-1.p} Sholander's basis for distributive lattices, part 5 (of 6).

This is part of the proof that Sholanders 2-basis for
 distributive lattices is correct. Here we prove associativity
 of join.




$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption},\mbox{axiom})$

$x\wedge ( y\vee z ) = ( z\wedge x )\vee ( y\wedge x )\qquad\mbox{cnf}(\mbox{distribution},\mbox{axiom})$

$( a\vee b )\vee c\ne a\vee ( b\vee c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_join},\mbox{negated\_conjecture})$

\medskip\textbf{LAT104-1.p} Huntington equation H3 is independent of H21

Show that Huntington equation H21 does not imply Huntington
 equation H3 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( a\wedge ( c\vee ( a\wedge b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT178-1.p} Equation H1 is Huntington by distributivity

Show that H1 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_1,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT014-1.p} McKenzie's 4-basis for lattice theory, part 3 (of 3)

This is part of a proof that McKenzie's 4-basis axiomatizes
 lattice theory.  We prove half of the standard basis.
 The other half follows by duality. In this part we prove
 absorbtion of meet.




$x\vee ( y\wedge ( x\wedge z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_1,\mbox{axiom})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_2,\mbox{axiom})$

$( ( x\wedge y )\vee ( y\wedge z ) )\vee y = y\qquad\mbox{cnf}(\mbox{mckenzie}_3,\mbox{axiom})$

$( ( x\vee y )\wedge ( y\vee z ) )\wedge y = y\qquad\mbox{cnf}(\mbox{mckenzie}_4,\mbox{axiom})$

$a\wedge ( a\vee b )\ne a\qquad\mbox{cnf}(\mbox{prove\_absorbtion},\mbox{negated\_conjecture})$

\medskip\textbf{LAT270-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_Aa1\_A\_58\_AA\_59\_Ab1\_A\_58\_AA\_59\_AS1\_A\_60\_61\_Ainterval\_Ar\_Aa1\_Ab1\_A\_124\_93\_A\_61\_61\_62\_AS1\_A\_60\_61\_AA\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

\medskip\textbf{LAT033-1.p} Idempotency of join






$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption}_1,\mbox{axiom})$

$x\vee ( x\wedge y ) = x\qquad\mbox{cnf}(\mbox{absorption}_2,\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( x\wedge y )\wedge z = x\wedge ( y\wedge z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet},\mbox{axiom})$

$( x\vee y )\vee z = x\vee ( y\vee z )\qquad\mbox{cnf}(\mbox{associativity\_of\_join},\mbox{axiom})$

$\mbox{xx}\vee \mbox{xx}\ne \mbox{xx}\qquad\mbox{cnf}(\mbox{idempotence\_of\_join},\mbox{negated\_conjecture})$

\medskip\textbf{LAT141-1.p} Huntington equation H21 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT211-1.p} Equation H69 is Huntington by distributivity

Show that H69 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge ( z\vee ( x\wedge y ) ) )\vee ( x\wedge ( y\vee ( x\wedge z ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{69},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT055-1.p} Lattice theory axioms






$\mbox{include}(\mbox{'Axioms/LAT002-0.ax'})$

\medskip\textbf{LAT250-1.p} Equation H68 is Huntington by implication

Show that H68 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{68},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT050-1.p} Orthomodular lattice is not modular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$x\vee ( x'\wedge ( x\vee y ) ) = x\vee y\qquad\mbox{cnf}(\mbox{orthomodular\_law},\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne ( a\vee b )\wedge ( a\vee c )\qquad\mbox{cnf}(\mbox{prove\_modular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT049-1.p} Ortholattice is not weakly orthomodular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$( a'\wedge ( a\vee b ) )\vee ( b'\vee ( a\wedge b ) )\ne n_1\qquad\mbox{cnf}(\mbox{prove\_weak\_orthomodular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT084-1.p} Axiom for lattice theory, part 5






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\vee b\ne b\vee a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LAT005-5.p} SAM's lemma

Let L be a modular lattice with 0 and 1.  Suppose that A and
 B are elements of L such that (A $\vee$ B) and (A $\wedge$ B) both have
 not necessarily unique complements. Then, (A $\vee$ B)' =
 ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ B)) $\wedge$ ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ A)).




$\mbox{include}(\mbox{'Axioms/LAT002-0.ax'})$

$a\wedge b{=}c\qquad\mbox{cnf}(\mbox{meet\_a\_and\_b},\mbox{negated\_conjecture})$

$c\vee r_2{=}n_1\qquad\mbox{cnf}(\mbox{join\_c\_and\_r}_2,\mbox{negated\_conjecture})$

$c\wedge r_2{=}n_0\qquad\mbox{cnf}(\mbox{meet\_c\_and\_r}_2,\mbox{negated\_conjecture})$

$r_2\wedge b{=}e\qquad\mbox{cnf}(\mbox{meet\_r2\_and\_b},\mbox{negated\_conjecture})$

$a\vee b{=}c_2\qquad\mbox{cnf}(\mbox{join\_a\_and\_b},\mbox{negated\_conjecture})$

$c_2\vee r_1{=}n_1\qquad\mbox{cnf}(\mbox{join\_c2\_and\_r}_1,\mbox{negated\_conjecture})$

$c_2\wedge r_1{=}n_0\qquad\mbox{cnf}(\mbox{meet\_c2\_and\_r}_1,\mbox{negated\_conjecture})$

$r_2\wedge a{=}d\qquad\mbox{cnf}(\mbox{meet\_r2\_and\_a},\mbox{negated\_conjecture})$

$r_1\vee e{=}a_2\qquad\mbox{cnf}(\mbox{join\_r1\_and\_e},\mbox{negated\_conjecture})$

$r_1\vee d{=}b_2\qquad\mbox{cnf}(\mbox{join\_r1\_and\_d},\mbox{negated\_conjecture})$

$\neg\,a_2\wedge b_2{=}r_1\qquad\mbox{cnf}(\mbox{meet\_a2\_and\_b}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT208-1.p} Equation H63 is Huntington by distributivity

Show that H63 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee z )\wedge ( u\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{63},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT162-1.p} Huntington equation H68 implies H73






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{68},\mbox{axiom})$

$a\wedge ( b\wedge ( c\vee d ) )\ne a\wedge ( b\wedge ( c\vee ( a\wedge ( d\vee ( b\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{73},\mbox{negated\_conjecture})$

\medskip\textbf{LAT085-1.p} Axiom for lattice theory, part 6






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a\vee b )\vee c\ne a\vee ( b\vee c )\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT006-1.p} Sholander's basis for distributive lattices, part 2 (of 6).

This is part of the proof that Sholanders 2-basis for
 distributive lattices is correct. Here we prove associativity
 of meet.




$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption},\mbox{axiom})$

$x\wedge ( y\vee z ) = ( z\wedge x )\vee ( y\wedge x )\qquad\mbox{cnf}(\mbox{distribution},\mbox{axiom})$

$( a\wedge b )\wedge c\ne a\wedge ( b\wedge c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT086-1.p} Axiom for lattice theory, part 7






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\wedge ( a\vee b )\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT261-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_in\_\_lattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT183-1.p} Equation H7 is Huntington by distributivity

Show that H7 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_7,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT028-1.p} Uniqueness of meet (dually join) in WAL

Let's say we have a weakly-associative lattice (WAL) with 2 meet
 operations, say meet1 and meet2.  In other words, {join,meet1}
 is a WAL, and {join,meet2} is a WAL.  Then, we can prove that the
 two meet operations are really the same.




$\mbox{include}(\mbox{'Axioms/LAT005-0.ax'})$

$x\wedge_2 x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet}_2,\mbox{axiom})$

$x\wedge_2 y = y\wedge_2 x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet}_2,\mbox{axiom})$

$( ( x\vee y )\wedge_2 ( z\vee y ) )\wedge_2 y = y\qquad\mbox{cnf}(\mbox{wal\_1}_2,\mbox{axiom})$

$( ( x\wedge_2 y )\vee ( z\wedge_2 y ) )\vee y = y\qquad\mbox{cnf}(\mbox{wal\_2}_2,\mbox{axiom})$

$a\wedge b\ne a\wedge_2 b\qquad\mbox{cnf}(\mbox{name},\mbox{negated\_conjecture})$

\medskip\textbf{LAT278-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\neg\,\mbox{c\_Relation\_Orefl}(\mbox{v\_A},\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LAT089-1.p} Absorption basis for WAL, part 2






$( a\wedge b )\vee ( a\wedge ( a\vee b ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_1,\mbox{axiom})$

$( a\wedge a )\vee ( b\wedge ( a\vee a ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_2,\mbox{axiom})$

$( a\wedge b )\vee ( b\wedge ( a\vee b ) ) = b\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_3,\mbox{axiom})$

$( ( a\vee b )\wedge ( c\vee a ) )\wedge a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_4,\mbox{axiom})$

$( ( a\wedge b )\vee ( c\wedge a ) )\vee a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_5,\mbox{axiom})$

$b\wedge a\ne a\wedge b\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT273-2.p} Problem about Tarski's fixed point theorem






$( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AS1\_A\_60\_61\_Ainterval\_Ar\_Aa1\_Ab1\_59\_Ax1\_A\_58\_AS1\_A\_124\_93\_A\_61\_61\_62\_A\_Ix1\_M\_Ab1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$( \mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_z},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_L},\mbox{v\_z},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{v\_sko\_\_4mi}(\mbox{v\_S},\mbox{v\_r},\mbox{v\_z}),\mbox{v\_S},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AisLub\_AS1\_Acl\_AL1\_59\_Az1\_A\_58\_AA\_59\_AALL\_Ay\_58S1\_O\_A\_Iy\_M\_Az1\_J\_A\_58\_Ar\_A\_124\_93\_A\_61\_61\_62\_A\_IL1\_M\_Az1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$( \mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_z},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_sko\_\_4mi}(\mbox{v\_S},\mbox{v\_r},\mbox{v\_z}),\mbox{v\_z},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_L},\mbox{v\_z},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AisLub\_AS1\_Acl\_AL1\_59\_Az1\_A\_58\_AA\_59\_AALL\_Ay\_58S1\_O\_A\_Iy\_M\_Az1\_J\_A\_58\_Ar\_A\_124\_93\_A\_61\_61\_62\_A\_IL1\_M\_Az1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_L},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT251-1.p} Equation H69 is Huntington by implication

Show that H69 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge ( z\vee ( x\wedge y ) ) )\vee ( x\wedge ( y\vee ( x\wedge z ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{69},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT252-1.p} Equation H70 is Huntington by implication

Show that H70 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( y\wedge ( x\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{70},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT073-1.p} Given single axiom MOL-23C, prove modularity

Given a single axiom candidate MOL-23C for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of modularity.




$f(f(f(b,f(a,b)),b),f(a,f(c,f(f(a,b),f(f(c,c),d))))) = a\qquad\mbox{cnf}(\mbox{mol\_23C},\mbox{axiom})$

$f(a,f(b,f(a,f(c,c))))\ne f(a,f(c,f(a,f(b,b))))\qquad\mbox{cnf}(\mbox{modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT271-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_S}\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT280-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\neg\,\mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LAT107-1.p} Huntington equation H17 is independent of H22

Show that Huntington equation H22 does not imply Huntington
 equation H17 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( z\vee ( x\wedge y ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{22},\mbox{axiom})$

$a\wedge ( ( a\wedge b )\vee ( a\wedge c ) )\ne a\wedge ( ( b\wedge ( a\vee ( b\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{17},\mbox{negated\_conjecture})$

\medskip\textbf{LAT277-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_Relation\_Orefl}(\mbox{v\_A},\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_OreflD}_0,\mbox{axiom})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT056-1.p} Lattice theory modularity (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

\medskip\textbf{LAT259-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_a}\ne \mbox{v\_b}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_U} = \mbox{v\_V}\qquad\mbox{cnf}(\mbox{cls\_Relation\_Oantisym\_\_def}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT181-1.p} Equation H4 is Huntington by distributivity

Show that H4 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\vee u ) ) ) = x\wedge ( y\vee ( ( x\vee ( y\wedge u ) )\wedge ( z\vee u ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_4,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT390-1.p} Short axiom for lattice theory, part 2






$( ( a + b )\cdot ( b + a\cdot b ) + c )\cdot ( ( a + ( ( d + b )\cdot ( b + e ) )\cdot b )\cdot ( ( b + ( ( d\cdot ( b\cdot e ) + f\cdot b ) + b ) )\cdot ( v_6 + b\cdot ( ( b\cdot v_7 + f\cdot b ) + b ) ) + a\cdot ( ( ( d + b )\cdot ( b + e ) )\cdot b ) ) + ( ( a + b )\cdot ( b + a\cdot b ) )\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$a + b\cdot ( a\cdot c )\ne a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT255-1.p} Equation H80 is Huntington by implication

Show that H80 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee ( y\wedge ( x\vee z ) ) ) ) ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{equation\_H}_{80},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT026-1.p} WAL + absorption gives LT, part 1.

A Weakly associative lattice (WAL) satisfying an absorption
 law is associative, and therefore a full lattice, part 1.




$\mbox{include}(\mbox{'Axioms/LAT005-0.ax'})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{absorbtion},\mbox{hypothesis})$

$( a\wedge b )\wedge c\ne a\wedge ( b\wedge c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT031-1.p} Distributivity of meet implies distributivity of join






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$\mbox{xx}\vee ( \mbox{yy}\wedge \mbox{zz} )\ne ( \mbox{xx}\vee \mbox{yy} )\wedge ( \mbox{xx}\vee \mbox{zz} )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{negated\_conjecture})$

\medskip\textbf{LAT145-1.p} Huntington equation H32 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\wedge u ) ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge u )\vee ( y\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{32},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT259-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_U} = \mbox{v\_V}\qquad\mbox{cnf}(\mbox{cls\_Relation\_Oantisym\_\_def}_0,\mbox{axiom})$

$\mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Main\_Oantisym\_\_def\_}_1(\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Main\_Oantisym\_\_def\_}_2(\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Oantisym\_\_def}_1,\mbox{axiom})$

$\mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Main\_Oantisym\_\_def\_}_2(\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Main\_Oantisym\_\_def\_}_1(\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Oantisym\_\_def}_2,\mbox{axiom})$

$\mbox{c\_Main\_Oantisym\_\_def\_}_1(\mbox{v\_r},\mbox{t\_a}) = \mbox{c\_Main\_Oantisym\_\_def\_}_2(\mbox{v\_r},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Relation\_Oantisym\_\_def}_3,\mbox{axiom})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_a}\ne \mbox{v\_b}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT186-1.p} Equation H11 is Huntington by distributivity

Show that H11 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{11},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT127-1.p} Huntington equation H6 is independent of H55\_dual

Show that Huntington equation H55\_dual does not imply Huntington
 equation H6 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H55\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT262-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_in\_\_lattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT076-1.p} Given single axiom MOL-27B1, prove associativity

Given a single axiom candidate MOL-27B1 for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of associativity.




$f(f(f(f(b,a),f(c,a)),d),f(a,f(f(f(f(f(f(b,b),a),c),c),a),b))) = a\qquad\mbox{cnf}(\mbox{mol\_27B}_1,\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT199-1.p} Equation H49 is Huntington by distributivity

Show that H49 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge ( y\vee u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{49},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT022-1.p} Self-dual modularity for quasilattices.






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$( x\wedge y )\vee ( z\wedge ( x\vee y ) ) = ( x\vee y )\wedge ( z\vee ( x\wedge y ) )\qquad\mbox{cnf}(\mbox{self\_dual\_modularity},\mbox{hypothesis})$

$a\wedge ( b\vee ( a\wedge c ) )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT185-1.p} Equation H10 is Huntington by distributivity

Show that H10 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{10},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT166-1.p} Huntington equation H77 implies H78






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{77},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( b\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( b\wedge ( a\vee d ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{78},\mbox{negated\_conjecture})$

\medskip\textbf{LAT005-6.p} SAM's lemma

Let L be a modular lattice with 0 and 1.  Suppose that A and
 B are elements of L such that (A $\vee$ B) and (A $\wedge$ B) both have
 not necessarily unique complements. Then, (A $\vee$ B)' =
 ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ B)) $\wedge$ ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ A)).




$\mbox{include}(\mbox{'Axioms/LAT002-0.ax'})$

$x\vee n_1{=}n_1\qquad\mbox{cnf}(\mbox{join\_x\_and}_1,\mbox{axiom})$

$x\vee n_0{=}x\qquad\mbox{cnf}(\mbox{join\_x\_and}_0,\mbox{axiom})$

$x\wedge n_0{=}n_0\qquad\mbox{cnf}(\mbox{meet\_x\_and}_0,\mbox{axiom})$

$x\wedge n_1{=}x\qquad\mbox{cnf}(\mbox{meet\_x\_and}_1,\mbox{axiom})$

$( x\wedge z{=}z\mbox{ and }y\vee z{=}y_1\mbox{ and }x\wedge y{=}x_1\mbox{ and }x\wedge y_1{=}z_1 ) \ \Rightarrow \ z\vee x_1{=}z_1\qquad\mbox{cnf}(\mbox{modularity}_3,\mbox{axiom})$

$( x\wedge z{=}z\mbox{ and }y\vee z{=}y_1\mbox{ and }x\wedge y{=}x_1\mbox{ and }z\vee x_1{=}z_1 ) \ \Rightarrow \ x\wedge y_1{=}z_1\qquad\mbox{cnf}(\mbox{modularity}_4,\mbox{axiom})$

$a\wedge b{=}c\qquad\mbox{cnf}(\mbox{meet\_a\_and\_b},\mbox{negated\_conjecture})$

$c\vee r_2{=}n_1\qquad\mbox{cnf}(\mbox{join\_c\_and\_r}_2,\mbox{negated\_conjecture})$

$c\wedge r_2{=}n_0\qquad\mbox{cnf}(\mbox{meet\_c\_and\_r}_2,\mbox{negated\_conjecture})$

$r_2\wedge b{=}e\qquad\mbox{cnf}(\mbox{meet\_r2\_and\_b},\mbox{negated\_conjecture})$

$a\vee b{=}c_2\qquad\mbox{cnf}(\mbox{join\_a\_and\_b},\mbox{negated\_conjecture})$

$c_2\vee r_1{=}n_1\qquad\mbox{cnf}(\mbox{join\_c2\_and\_r}_1,\mbox{negated\_conjecture})$

$c_2\wedge r_1{=}n_0\qquad\mbox{cnf}(\mbox{meet\_c2\_and\_r}_1,\mbox{negated\_conjecture})$

$r_2\wedge a{=}d\qquad\mbox{cnf}(\mbox{meet\_r2\_and\_a},\mbox{negated\_conjecture})$

$r_1\vee e{=}a_2\qquad\mbox{cnf}(\mbox{join\_r1\_and\_e},\mbox{negated\_conjecture})$

$r_1\vee d{=}b_2\qquad\mbox{cnf}(\mbox{join\_r1\_and\_d},\mbox{negated\_conjecture})$

$\neg\,a_2\wedge b_2{=}r_1\qquad\mbox{cnf}(\mbox{meet\_a2\_and\_b}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT045-1.p} Lattice orthomodular law from modular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$x\vee ( y\wedge ( x\vee z ) ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{modular\_law},\mbox{axiom})$

$a\vee ( a'\wedge ( a\vee b ) )\ne a\vee b\qquad\mbox{cnf}(\mbox{prove\_orthomodular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT266-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT066-1.p} Weak property G61 to make a uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( b\vee ( c\vee ( d\wedge ( a\vee ( b\wedge c ) ) ) ) ) = a\wedge ( b\vee ( c\vee ( a\wedge d ) ) )\qquad\mbox{cnf}(g_{61},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT071-1.p} Given single axiom OML-21C, prove associativity

Given a single axiom candidate OML-21C for orthomodular lattices
 (OML) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of associativity.




$f(f(b,a),f(f(f(f(b,a),a),f(c,a)),f(f(a,a),d))) = a\qquad\mbox{cnf}(\mbox{oml\_21C},\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT118-1.p} Huntington equation H69 is independent of H79

Show that Huntington equation H79 does not imply Huntington
 equation H69 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge u ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{79},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge ( c\vee ( a\wedge b ) ) )\vee ( a\wedge ( b\vee ( a\wedge c ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LAT143-1.p} Huntington equation H24 implies H15






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( y\wedge z ) = ( x\wedge y )\vee ( y\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{24},\mbox{axiom})$

$a\wedge ( ( a\wedge b )\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge b )\vee ( ( a\wedge c )\vee ( c\wedge ( a\vee b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{15},\mbox{negated\_conjecture})$

\medskip\textbf{LAT075-1.p} Given single axiom MOL-25A, prove modularity

Given a single axiom candidate MOL-25A for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of modularity.




$f(f(b,a),f(f(f(a,a),c),f(f(f(f(f(a,b),c),c),a),f(a,d)))) = a\qquad\mbox{cnf}(\mbox{mol\_25A},\mbox{axiom})$

$f(a,f(b,f(a,f(c,c))))\ne f(a,f(c,f(a,f(b,b))))\qquad\mbox{cnf}(\mbox{modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT063-1.p} E62 does not necessarily hold in ortholattices






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a'\vee a = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a'\wedge a = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$a\wedge b = ( a'\vee b' )'\qquad\mbox{cnf}(\mbox{compatibility},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( a'\vee ( a\wedge b ) ) ) )\ne a\wedge ( a'\vee ( a\wedge b ) )\qquad\mbox{cnf}(\mbox{prove\_e}_{62},\mbox{negated\_conjecture})$

\medskip\textbf{LAT046-1.p} Modular ortholattice is not Boolean algebra






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$x\vee ( y\wedge ( x\vee z ) ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{modular\_law},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT198-1.p} Equation H42 is Huntington by distributivity

Show that H42 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\vee ( x\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{42},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT157-1.p} Huntington equation H50 implies H2






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge ( b\vee c ) )\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT021-1.p} Bowden's inequality gives distributivity in lattice theory.






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$( x\vee ( y\wedge z ) )\vee ( ( x\vee y )\wedge z ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{bowden},\mbox{hypothesis})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT217-1.p} Equation H82 is Huntington by distributivity

Show that H82 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\wedge ( x\vee z ) )\vee ( z\wedge ( x\vee y ) ) ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{equation\_H}_{82},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT074-1.p} Given single axiom MOL-25A, prove associativity

Given a single axiom candidate MOL-25A for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of associativity.




$f(f(b,a),f(f(f(a,a),c),f(f(f(f(f(a,b),c),c),a),f(a,d)))) = a\qquad\mbox{cnf}(\mbox{mol\_25A},\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT223-1.p} Equation H7 is Huntington by implication

Show that H7 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_7,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT204-1.p} Equation H58 is Huntington by distributivity

Show that H58 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( ( x\vee y )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{58},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT048-1.p} Weakly orthomodular lattice is not orthomodular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$( x'\wedge ( x\vee y ) )\vee ( y'\vee ( x\wedge y ) ) = n_1\qquad\mbox{cnf}(\mbox{weak\_orthomodular\_law},\mbox{hypothesis})$

$a\vee ( a'\wedge ( a\vee b ) )\ne a\vee b\qquad\mbox{cnf}(\mbox{prove\_orthomodular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT012-1.p} McKenzie's 4-basis for lattice theory, part 1 (of 3)

This is part of a proof that McKenzie's 4-basis axiomatizes
 lattice theory.  We prove half of the standard basis.
 The other half follows by duality.  In this part we prove
 commutativity of meet.




$x\vee ( y\wedge ( x\wedge z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_1,\mbox{axiom})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_2,\mbox{axiom})$

$( ( x\wedge y )\vee ( y\wedge z ) )\vee y = y\qquad\mbox{cnf}(\mbox{mckenzie}_3,\mbox{axiom})$

$( ( x\vee y )\wedge ( y\vee z ) )\wedge y = y\qquad\mbox{cnf}(\mbox{mckenzie}_4,\mbox{axiom})$

$b\wedge a\ne a\wedge b\qquad\mbox{cnf}(\mbox{prove\_commutativity\_of\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT274-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_S}\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_L},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT139-1.p} Huntington equation H11 implies H10






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{11},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( a\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{LAT233-1.p} Equation H24 is Huntington by implication

Show that H24 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( y\wedge z ) = ( x\wedge y )\vee ( y\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{24},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT269-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OTop\_Acl\_A\_58\_AA\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{c\_Tarski\_Oinduced}(\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Product\_\_Type\_OUnity},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ or }\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_emptyset} )\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_Aa2\_A\_58\_AA\_59\_Ab2\_A\_58\_AA\_59\_Ainterval\_Ar\_Aa2\_Ab2\_A\_126\_61\_A\_123\_125\_A\_124\_93\_A\_61\_61\_62\_A\_I\_124\_Apset\_A\_61\_Ainterval\_Ar\_Aa2\_Ab2\_M\_Aorder\_A\_61\_Ainduced\_A\_Iinterval\_Ar\_Aa2\_Ab2\_J\_Ar\_A\_124\_J\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{v\_intY}_1 = \mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{c\_Tarski\_Olub}(\mbox{v\_Y},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OintY1\_A\_61\_61\_Ainterval\_Ar\_A\_Ilub\_AY\_Acl\_J\_A\_ITop\_Acl\_J}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_Y},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Olub\_AY\_Acl\_A\_58\_AA\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_x},\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ox1\_A\_58\_AA\_A\_61\_61\_62\_Ainterval\_Ar\_Ax1\_A\_ITop\_Acl\_J\_A\_61\_A\_123\_125\_A\_61\_61\_AFalse}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_intY}_1,\mbox{c\_Tarski\_Oinduced}(\mbox{v\_intY}_1,\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Product\_\_Type\_OUnity},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LAT232-1.p} Equation H22 is Huntington by implication

Show that H22 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( z\vee ( x\wedge y ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{22},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT013-1.p} McKenzie's 4-basis for lattice theory, part 2 (of 3)

This is part of a proof that McKenzie's 4-basis axiomatizes
 lattice theory.  We prove half of the standard basis.
 The other half follows by duality.  In this part we prove
 associativity of meet.




$x\vee ( y\wedge ( x\wedge z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_1,\mbox{axiom})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{mckenzie}_2,\mbox{axiom})$

$( ( x\wedge y )\vee ( y\wedge z ) )\vee y = y\qquad\mbox{cnf}(\mbox{mckenzie}_3,\mbox{axiom})$

$( ( x\vee y )\wedge ( y\vee z ) )\wedge y = y\qquad\mbox{cnf}(\mbox{mckenzie}_4,\mbox{axiom})$

$( a\wedge b )\wedge c\ne a\wedge ( b\wedge c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT176-1.p} Huntington equation H79\_dual implies H42






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\vee ( ( x\vee ( y\wedge ( x\vee z ) ) )\wedge ( z\vee u ) )\qquad\mbox{cnf}(\mbox{equation\_H79\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT123-1.p} Huntington equation H55 is independent of H22\_dual

Show that Huntington equation H22\_dual does not imply Huntington
 equation H55 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( y\vee ( z\wedge ( x\vee y ) ) )\wedge ( z\vee ( x\wedge y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H22\_dual},\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne a\vee ( b\wedge ( c\vee ( a\wedge ( c\vee b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{55},\mbox{negated\_conjecture})$

\medskip\textbf{LAT130-1.p} Huntington equation H39 is independent of H68\_dual

Show that Huntington equation H68\_dual does not imply Huntington
 equation H39 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = x\vee ( y\wedge ( x\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H68\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( a\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LAT280-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT129-1.p} Huntington equation H10 is independent of H58\_dual

Show that Huntington equation H58\_dual does not imply Huntington
 equation H10 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = x\vee ( y\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H58\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( a\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{LAT002-1.p} If X' = U $\vee$ V and Y' = U $\wedge$ V, then U' exists

The theorem states that there is a complement of "a" in a
 modular lattice with 0 and 1.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$r_1'{=}a\vee b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_join\_b},\mbox{hypothesis})$

$r_2'{=}a\wedge b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_meet\_b},\mbox{hypothesis})$

$\neg\,a'{=}w\qquad\mbox{cnf}(\mbox{prove\_complememt\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{LAT090-1.p} Absorption basis for WAL, part 3






$( a\wedge b )\vee ( a\wedge ( a\vee b ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_1,\mbox{axiom})$

$( a\wedge a )\vee ( b\wedge ( a\vee a ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_2,\mbox{axiom})$

$( a\wedge b )\vee ( b\wedge ( a\vee b ) ) = b\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_3,\mbox{axiom})$

$( ( a\vee b )\wedge ( c\vee a ) )\wedge a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_4,\mbox{axiom})$

$( ( a\wedge b )\vee ( c\wedge a ) )\vee a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_5,\mbox{axiom})$

$a\vee a\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT131-1.p} Huntington equation H42 is independent of H68\_dual

Show that Huntington equation H68\_dual does not imply Huntington
 equation H42 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = x\vee ( y\wedge ( x\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H68\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT156-1.p} Huntington equation H49 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge ( y\vee u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{49},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT081-1.p} Axiom for lattice theory, part 2






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\wedge b\ne b\wedge a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT043-1.p} Lattice compatability from Boolean algebra






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{distributivity},\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$( c\vee d )'\ne c'\wedge d'\qquad\mbox{cnf}(\mbox{prove\_compatability\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT214-1.p} Equation H79 is Huntington by distributivity

Show that H79 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge u ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{79},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT059-1.p} Ortholattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT003-0.ax'})$

\medskip\textbf{LAT149-1.p} Huntington equation H37 implies H43






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\vee ( x\wedge u ) ) ) = x\wedge ( y\vee ( z\vee ( u\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{37},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( b\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( a\wedge ( b\vee d ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{43},\mbox{negated\_conjecture})$

\medskip\textbf{LAT271-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AS1\_A\_60\_61\_Ainterval\_Ar\_Aa1\_Ab1\_59\_Ax1\_A\_58\_AS1\_A\_124\_93\_A\_61\_61\_62\_A\_Ia1\_M\_Ax1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

\medskip\textbf{LAT264-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

\medskip\textbf{LAT197-1.p} Equation H40 is Huntington by distributivity

Show that H40 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{40},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT278-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_Relation\_Orefl}(\mbox{v\_A},\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT058-1.p} Lattice theory unique complement (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-3.ax'})$

\medskip\textbf{LAT110-1.p} Huntington equation H42 is independent of H37

Show that Huntington equation H37 does not imply Huntington
 equation H42 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\vee ( x\wedge u ) ) ) = x\wedge ( y\vee ( z\vee ( u\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{37},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT249-1.p} Equation H64 is Huntington by implication

Show that H64 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge ( y\vee ( x\wedge z ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{64},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT158-1.p} Huntington equation H50 implies H49






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( ( a\wedge c )\vee ( c\wedge ( b\vee d ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{49},\mbox{negated\_conjecture})$

\medskip\textbf{LAT037-1.p} Uniqueness of complement

Distributive lattice complements are unique whenever they exist.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$x\wedge n_0 = n_0\qquad\mbox{cnf}(\mbox{x\_meet}_0,\mbox{axiom})$

$x\vee n_0 = x\qquad\mbox{cnf}(\mbox{x\_join}_0,\mbox{axiom})$

$x\wedge n_1 = x\qquad\mbox{cnf}(\mbox{x\_meet}_1,\mbox{axiom})$

$x\vee n_1 = n_1\qquad\mbox{cnf}(\mbox{x\_join}_1,\mbox{axiom})$

$x\wedge z = x \ \Rightarrow \ z\wedge ( x\vee y ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{modular},\mbox{axiom})$

$\mbox{xx}\vee \mbox{yy} = n_1\qquad\mbox{cnf}(\mbox{lhs}_1,\mbox{axiom})$

$\mbox{xx}\vee \mbox{zz} = n_1\qquad\mbox{cnf}(\mbox{lhs}_2,\mbox{axiom})$

$\mbox{xx}\wedge \mbox{yy} = n_0\qquad\mbox{cnf}(\mbox{lhs}_3,\mbox{axiom})$

$\mbox{xx}\wedge \mbox{zz} = n_0\qquad\mbox{cnf}(\mbox{lhs}_4,\mbox{axiom})$

$\mbox{yy}\ne \mbox{zz}\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT126-1.p} Huntington equation H69 is independent of H39\_dual

Show that Huntington equation H39\_dual does not imply Huntington
 equation H69 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\vee ( y\wedge ( z\vee ( u\wedge ( x\vee z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H39\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge ( c\vee ( a\wedge b ) ) )\vee ( a\wedge ( b\vee ( a\wedge c ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LAT055-2.p} Lattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

\medskip\textbf{LAT160-1.p} Huntington equation H52 implies H51






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( z\wedge u )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{52},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( ( a\wedge c )\vee ( c\wedge d ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{51},\mbox{negated\_conjecture})$

\medskip\textbf{LAT216-1.p} Equation H81 is Huntington by distributivity

Show that H81 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee ( y\wedge ( x\vee z ) ) ) ) ) = x\wedge ( ( x\wedge y )\vee ( x\wedge z ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{81},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT097-1.p} Single axiom for weakly associative lattices (WAL), part 6






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( ( a\wedge b )\vee ( c\wedge b ) )\vee b\ne b\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT087-1.p} Axiom for lattice theory, part 8






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\vee ( a\wedge b )\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_8,\mbox{negated\_conjecture})$

\medskip\textbf{LAT277-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_Relation\_Orefl}(\mbox{v\_A},\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_OreflD}_0,\mbox{axiom})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT381+1.p} Tarski-Knaster fixed point theorem 01, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isEmpty}_0(w_0)\iff \neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmpty},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtlseqdt}_0(w_0,w_1) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mLessRel},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mARefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{sdtlseqdt}_0(w_0,w_1)\mbox{ and }\mbox{sdtlseqdt}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1)\mbox{ and }\mbox{aElement}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{sdtlseqdt}_0(w_0,w_1)\mbox{ and }\mbox{sdtlseqdt}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mTrans},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aLowerBoundOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_1) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_2,w_3) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefLB},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aUpperBoundOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_1) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_3,w_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefUB},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aInfimumOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\mbox{aLowerBoundOfIn}_0(w_2,w_1,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aLowerBoundOfIn}_0(w_3,w_1,w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_3,w_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefInf},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aSupremumOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\mbox{aUpperBoundOfIn}_0(w_2,w_1,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aUpperBoundOfIn}_0(w_3,w_1,w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_2,w_3) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSup},\mbox{definition})$

$\mbox{aSet}_0(\mbox{xT})\qquad\mbox{fof}(\mbox{m\_}_{725},\mbox{hypothesis})$

$\mbox{aSubsetOf}_0(\mbox{xS},\mbox{xT})\qquad\mbox{fof}(\mbox{m\_\_725}_{01},\mbox{hypothesis})$

$\mbox{aSupremumOfIn}_0(\mbox{xu},\mbox{xS},\mbox{xT})\mbox{ and }\mbox{aSupremumOfIn}_0(\mbox{xv},\mbox{xS},\mbox{xT})\qquad\mbox{fof}(\mbox{m\_}_{744},\mbox{hypothesis})$

$\mbox{xu} = \mbox{xv}\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{LAT267-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT201-1.p} Equation H51 is Huntington by distributivity

Show that H51 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge u ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{51},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT082-1.p} Axiom for lattice theory, part 3






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a\wedge b )\wedge c\ne a\wedge ( b\wedge c )\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT268-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT088-1.p} Absorption basis for WAL, part 1






$( a\wedge b )\vee ( a\wedge ( a\vee b ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_1,\mbox{axiom})$

$( a\wedge a )\vee ( b\wedge ( a\vee a ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_2,\mbox{axiom})$

$( a\wedge b )\vee ( b\wedge ( a\vee b ) ) = b\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_3,\mbox{axiom})$

$( ( a\vee b )\wedge ( c\vee a ) )\wedge a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_4,\mbox{axiom})$

$( ( a\wedge b )\vee ( c\wedge a ) )\vee a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_5,\mbox{axiom})$

$a\wedge a\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT159-1.p} Huntington equation H50 implies H7






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( a\wedge ( ( a\wedge b )\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT117-1.p} Huntington equation H69 is independent of H65

Show that Huntington equation H65 does not imply Huntington
 equation H69 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{65},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge ( c\vee ( a\wedge b ) ) )\vee ( a\wedge ( b\vee ( a\wedge c ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LAT114-1.p} Huntington equation H56 is independent of H55

Show that Huntington equation H55 does not imply Huntington
 equation H56 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$( a\wedge b )\vee ( a\wedge ( b\vee c ) )\ne a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{56},\mbox{negated\_conjecture})$

\medskip\textbf{LAT078-1.p} Given single axiom MOL-27B2, prove associativity

Given a single axiom candidate MOL-27B2 for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of associativity.




$f(f(f(f(b,a),f(a,c)),d),f(a,f(f(f(b,f(b,f(f(c,c),a))),a),c))) = a\qquad\mbox{cnf}(\mbox{mol\_27B}_2,\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT224-1.p} Equation H8 is Huntington by implication

Show that H8 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = ( x\wedge y )\vee ( x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_8,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT093-1.p} Axiom for weakly associative lattices (WAL), part 2






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$b\wedge a\ne a\wedge b\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT229-1.p} Equation H17 is Huntington by implication

Show that H17 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{17},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT179-1.p} Equation H2 is Huntington by distributivity

Show that H2 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge ( y\vee z ) )\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_2,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT064-1.p} Weak property 94-6 to make a uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee ( b\wedge c ) ) ) ) = a\wedge ( b\vee ( a\wedge c ) )\qquad\mbox{cnf}(\mbox{c94}_6,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT032-1.p} Distributivity of join implies distributivity of meet






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$\mbox{xx}\wedge ( \mbox{yy}\vee \mbox{zz} )\ne ( \mbox{xx}\wedge \mbox{yy} )\vee ( \mbox{xx}\wedge \mbox{zz} )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT393-2.p} Ortholattices in terms of Sheffer stroke: associativity






$f(f(f(f(a,b),f(b,c)),d),f(b,f(f(b,f(f(a,a),a)),c))) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$f(x_0,f(f(x_1,x_2),f(x_1,x_2)))\ne f(x_1,f(f(x_0,x_2),f(x_0,x_2)))\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT164-1.p} Huntington equation H76 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{76},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT039-2.p} Every distributive lattice is modular

Theorem formulation : Modularity is expressed by:
 x $\leq$ y $\to$ x $\vee$ (y \& z) = (x $\vee$ y) \& (x $\vee$ z)




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$\mbox{xx}\vee \mbox{yy} = \mbox{yy}\qquad\mbox{cnf}(\mbox{lhs},\mbox{hypothesis})$

$\mbox{xx}\vee ( \mbox{yy}\wedge \mbox{zz} )\ne ( \mbox{xx}\vee \mbox{yy} )\wedge ( \mbox{xx}\vee \mbox{zz} )\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT394-2.p} Ortholattices in terms of Sheffer stroke: unit






$f(f(f(f(a,b),f(b,c)),d),f(b,f(f(b,f(f(a,a),a)),c))) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$f(x_0,f(x_0,x_0))\ne f(x_1,f(x_1,x_1))\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT010-1.p} McKenzie's basis for the variety generated by N5.

McKenzie's basis for the variety generated by N5.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge u ) ) )\qquad\mbox{cnf}(\mbox{n5}_1,\mbox{hypothesis})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = ( x\vee ( y\wedge ( x\vee z ) ) )\wedge ( x\vee ( ( x\vee y )\wedge ( z\vee u ) ) )\qquad\mbox{cnf}(\mbox{n5}_2,\mbox{hypothesis})$

$( x\vee ( y\wedge z ) )\wedge ( z\vee ( x\wedge y ) ) = ( z\wedge ( x\vee ( y\wedge z ) ) )\vee ( x\wedge ( y\vee z ) )\qquad\mbox{cnf}(\mbox{nr}_3,\mbox{hypothesis})$

$a\wedge ( ( b\vee c )\wedge ( b\vee d ) )\ne ( a\wedge ( ( b\vee c )\wedge ( b\vee d ) ) )\wedge ( ( a\wedge ( b\vee ( c\wedge d ) ) )\vee ( ( a\wedge c )\vee ( a\wedge d ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LAT146-1.p} Huntington equation H34 implies H28






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\wedge ( y\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{34},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\wedge d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\wedge ( a\vee ( b\wedge d ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{28},\mbox{negated\_conjecture})$

\medskip\textbf{LAT242-1.p} Equation H55 is Huntington by implication

Show that H55 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT147-1.p} Huntington equation H34 implies H45






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\wedge ( y\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{34},\mbox{axiom})$

$a\wedge ( b\wedge ( c\vee ( a\wedge d ) ) )\ne a\wedge ( b\wedge ( c\vee ( d\wedge ( a\vee ( b\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{45},\mbox{negated\_conjecture})$

\medskip\textbf{LAT239-1.p} Equation H49 is Huntington by implication

Show that H49 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge ( y\vee u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{49},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT202-1.p} Equation H55 is Huntington by distributivity

Show that H55 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT152-1.p} Huntington equation H40 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{40},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT040-1.p} Another simplification rule for distributive lattices

In every distributive lattice the simplification rule holds:
 forall x, y, z: (x $\vee$ y = x $\vee$ z, x \& y = x \& z $\to$ y = z ).




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$\mbox{xx}\vee \mbox{yy} = \mbox{xx}\vee \mbox{zz}\qquad\mbox{cnf}(\mbox{lhs}_1,\mbox{hypothesis})$

$\mbox{xx}\wedge \mbox{yy} = \mbox{xx}\wedge \mbox{zz}\qquad\mbox{cnf}(\mbox{lhs}_2,\mbox{hypothesis})$

$\mbox{yy}\ne \mbox{zz}\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT262-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT065-1.p} Weak property 94-37 to make uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( ( b\wedge ( c\vee ( a\wedge b ) ) )\vee ( c\wedge ( a\vee b ) ) ) = ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{c94}_{37},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT180-1.p} Equation H3 is Huntington by distributivity

Show that H3 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_3,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT044-1.p} Lattice weak orthomodular law from orthomodular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$x\vee ( x'\wedge ( x\vee y ) ) = x\vee y\qquad\mbox{cnf}(\mbox{orthomodular\_law},\mbox{axiom})$

$( a'\wedge ( a\vee b ) )\vee ( b'\vee ( a\wedge b ) )\ne n_1\qquad\mbox{cnf}(\mbox{prove\_weak\_orthomodular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT051-1.p} Lattice is not ortholattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$( a\vee b )'\ne a'\wedge b'\qquad\mbox{cnf}(\mbox{prove\_compatibility\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT041-1.p} A hyperbase for type $<$2,2$>$ lattice hyperidentities






$\mbox{big\_p}(\mbox{term}_1,x,y) = x\qquad\mbox{cnf}(\mbox{big\_p\_term}_1,\mbox{axiom})$

$\mbox{big\_p}(\mbox{term}_2,x,y) = y\qquad\mbox{cnf}(\mbox{big\_p\_term}_2,\mbox{axiom})$

$\mbox{big\_p}(\mbox{term}_3,x,y) = \mbox{times}(x,y)\qquad\mbox{cnf}(\mbox{big\_p\_term}_3,\mbox{axiom})$

$\mbox{big\_p}(\mbox{term}_4,x,y) = \mbox{times}(y,x)\qquad\mbox{cnf}(\mbox{big\_p\_term}_4,\mbox{axiom})$

$\mbox{big\_p}(\mbox{term}_5,x,y) = x + y\qquad\mbox{cnf}(\mbox{big\_p\_term}_5,\mbox{axiom})$

$\mbox{big\_p}(\mbox{term}_6,x,y) = y + x\qquad\mbox{cnf}(\mbox{big\_p\_term}_6,\mbox{axiom})$

$\mbox{big\_t}(w,x,y) = \mbox{big\_p}(w,x,y)\qquad\mbox{cnf}(\mbox{big\_p\_and\_big\_t},\mbox{axiom})$

$\mbox{term}(\mbox{term}_1)\qquad\mbox{cnf}(\mbox{term}_1,\mbox{axiom})$

$\mbox{term}(\mbox{term}_2)\qquad\mbox{cnf}(\mbox{term}_2,\mbox{axiom})$

$\mbox{term}(\mbox{term}_3)\qquad\mbox{cnf}(\mbox{term}_3,\mbox{axiom})$

$\mbox{term}(\mbox{term}_4)\qquad\mbox{cnf}(\mbox{term}_4,\mbox{axiom})$

$\mbox{term}(\mbox{term}_5)\qquad\mbox{cnf}(\mbox{term}_5,\mbox{axiom})$

$\mbox{term}(\mbox{term}_6)\qquad\mbox{cnf}(\mbox{term}_6,\mbox{axiom})$

$\mbox{term}(w) \ \Rightarrow \ \mbox{big\_p}(w,\mbox{big\_p}(w,x,y),z) = \mbox{big\_p}(w,x,\mbox{big\_p}(w,y,z))\qquad\mbox{cnf}(q_2,\mbox{hypothesis})$

$\mbox{term}(w) \ \Rightarrow \ \mbox{big\_p}(w,\mbox{big\_p}(w,x,y),\mbox{big\_p}(w,z,v)) = \mbox{big\_p}(w,\mbox{big\_p}(w,x,z),\mbox{big\_p}(w,y,v))\qquad\mbox{cnf}(q_5,\mbox{hypothesis})$

$( \mbox{term}(w_1)\mbox{ and }\mbox{term}(w_2) ) \ \Rightarrow \ \mbox{big\_t}(w_1,\mbox{big\_p}(w_2,\mbox{big\_t}(w_1,x,y),z),\mbox{big\_p}(w_2,y,z)) = \mbox{big\_p}(w_2,\mbox{big\_t}(w_1,x,y),z)\qquad\mbox{cnf}(q_3,\mbox{hypothesis})$

$\mbox{times}(a,b + \mbox{times}(d,\mbox{times}(c,e)))\ne \mbox{times}(a,b + \mbox{times}(b + c,\mbox{times}(d,\mbox{times}(c,e))))\qquad\mbox{cnf}(\mbox{prove\_q}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT100-1.p} Huntington equation H4 is independent of H6

Show that Huntington equation H6 does not imply Huntington
 equation H4 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_6,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\vee d ) ) )\ne a\wedge ( b\vee ( ( a\vee ( b\wedge d ) )\wedge ( c\vee d ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT140-1.p} Huntington equation H21 implies H2






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge ( b\vee c ) )\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT053-1.p} Countermodel for Megill equation for weakly orthomodular lattices






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$( x'\wedge ( x\vee y ) )\vee ( y'\vee ( x\wedge y ) ) = n_1\qquad\mbox{cnf}(\mbox{megill},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( a'\vee ( a\wedge b ) ) ) )\ne a\wedge ( a'\vee ( a\wedge b ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LAT099-1.p} Huntington equation H2 is independent of H3

Show that Huntington equation H3 does not imply Huntington
 equation H2 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_3,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge ( b\vee c ) )\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT392-1.p} Short axiom for lattice theory, part 4






$( ( a + b )\cdot ( b + a\cdot b ) + c )\cdot ( ( a + ( ( d + b )\cdot ( b + e ) )\cdot b )\cdot ( ( b + ( ( d\cdot ( b\cdot e ) + f\cdot b ) + b ) )\cdot ( v_6 + b\cdot ( ( b\cdot v_7 + f\cdot b ) + b ) ) + a\cdot ( ( ( d + b )\cdot ( b + e ) )\cdot b ) ) + ( ( a + b )\cdot ( b + a\cdot b ) )\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$( a\cdot b + b\cdot c ) + b\ne b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT261-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

\medskip\textbf{LAT001-1.p} If X' = U $\vee$ V and Y' = U $\wedge$ V, then U' = X $\vee$ (Y $\wedge$ V)

The theorem states that there is a complement of "a" in a
 modular lattice with 0 and 1.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$r_1'{=}a\vee b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_join\_b},\mbox{hypothesis})$

$r_2'{=}a\wedge b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_meet\_b},\mbox{hypothesis})$

$\neg\,a'{=}r_1\vee ( r_2\wedge b )\qquad\mbox{cnf}(\mbox{prove\_complememt},\mbox{negated\_conjecture})$

\medskip\textbf{LAT039-1.p} Every distributive lattice is modular






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$\mbox{xx}\vee \mbox{yy} = \mbox{yy}\qquad\mbox{cnf}(\mbox{lhs},\mbox{hypothesis})$

$\mbox{xx}\vee ( \mbox{yy}\wedge \mbox{zz} )\ne \mbox{yy}\wedge ( \mbox{xx}\vee \mbox{zz} )\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT111-1.p} Huntington equation H40 is independent of H45

Show that Huntington equation H45 does not imply Huntington
 equation H40 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\wedge ( y\wedge ( z\vee ( u\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{45},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LAT027-1.p} WAL + absorption gives LT, part 2.

A Weakly associative lattice (WAL) satisfying an absorption
 law is associative, and therefore a full lattice, part 2.




$\mbox{include}(\mbox{'Axioms/LAT005-0.ax'})$

$x\wedge ( y\vee ( x\vee z ) ) = x\qquad\mbox{cnf}(\mbox{absorption},\mbox{hypothesis})$

$( a\vee b )\vee c\ne a\vee ( b\vee c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_join},\mbox{negated\_conjecture})$

\medskip\textbf{LAT016-1.p} E1 fails for Ortholattices.

Show that Ortholattices do not necessarily satisfy equation E1.




$\mbox{include}(\mbox{'Axioms/LAT003-0.ax'})$

$( ( a\wedge b' )\vee a' )'\vee ( ( a\wedge b' )\vee ( ( a'\wedge ( ( a\vee b' )\wedge ( a\vee b ) ) )\vee ( a'\wedge ( ( a\vee b' )\wedge ( a\vee b ) )' ) ) )\ne n_1\qquad\mbox{cnf}(\mbox{prove\_e}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT236-1.p} Equation H39 is Huntington by implication

Show that H39 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{39},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT124-1.p} Huntington equation H69 is independent of H32\_dual

Show that Huntington equation H32\_dual does not imply Huntington
 equation H69 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee ( z\vee u ) ) ) = x\vee ( y\wedge ( z\vee ( ( x\vee u )\wedge ( y\vee u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H32\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge ( c\vee ( a\wedge b ) ) )\vee ( a\wedge ( b\vee ( a\wedge c ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LAT172-1.p} Huntington equation H76\_dual implies H32






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( y\wedge u ) ) ) = x\vee ( y\wedge ( z\vee ( u\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H76\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\wedge d ) ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge d )\vee ( b\wedge d ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{LAT069-1.p} Weak property G113 to make a uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( ( a\wedge c )\vee ( d\wedge ( b\vee c ) ) ) ) ) = a\wedge ( b\vee ( ( a\wedge c )\vee ( c\wedge d ) ) )\qquad\mbox{cnf}(g_{113},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT042-1.p} Lattice modularity from Boolean algebra






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{distributivity},\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne ( a\vee b )\wedge ( a\vee c )\qquad\mbox{cnf}(\mbox{prove\_modular\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT221-1.p} Equation H4 is Huntington by implication

Show that H4 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\vee u ) ) ) = x\wedge ( y\vee ( ( x\vee ( y\wedge u ) )\wedge ( z\vee u ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_4,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT060-1.p} Quasilattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

\medskip\textbf{LAT091-1.p} Absorption basis for WAL, part 4






$( a\wedge b )\vee ( a\wedge ( a\vee b ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_1,\mbox{axiom})$

$( a\wedge a )\vee ( b\wedge ( a\vee a ) ) = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_2,\mbox{axiom})$

$( a\wedge b )\vee ( b\wedge ( a\vee b ) ) = b\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_3,\mbox{axiom})$

$( ( a\vee b )\wedge ( c\vee a ) )\wedge a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_4,\mbox{axiom})$

$( ( a\wedge b )\vee ( c\wedge a ) )\vee a = a\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_5,\mbox{axiom})$

$b\vee a\ne a\vee b\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT036-1.p} Property of a distributive lattive with an antimorphism

In every distributive lattice with 0,1 and an antimorphism k:
 if   k$\wedge$2(a) $\leq$ a $\vee$ k(a)              and
 k$\wedge$3(b) $\leq$ a $\vee$ k(a)              and
 k$\wedge$2(a) $\leq$ k(a) $\vee$ k(b) $\vee$ k(c)    and
 k$\wedge$3(b) $\leq$ k(a) $\vee$ k(b) $\vee$ k(c)
 then k$\wedge$2(a $\vee$ k(b)) $\leq$ (a \& k(b \& c)) $\vee$ k(a)




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$x\wedge n_0 = n_0\qquad\mbox{cnf}(\mbox{x\_meet}_0,\mbox{axiom})$

$x\vee n_0 = x\qquad\mbox{cnf}(\mbox{x\_join}_0,\mbox{axiom})$

$x\wedge n_1 = x\qquad\mbox{cnf}(\mbox{x\_meet}_1,\mbox{axiom})$

$x\vee n_1 = n_1\qquad\mbox{cnf}(\mbox{x\_join}_1,\mbox{axiom})$

$x\wedge z = x \ \Rightarrow \ z\wedge ( x\vee y ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{modular},\mbox{axiom})$

$k(u\vee v) = k(u)\wedge k(v)\qquad\mbox{cnf}(\mbox{k\_on\_join},\mbox{axiom})$

$k(u\wedge v) = k(u)\vee k(v)\qquad\mbox{cnf}(\mbox{k\_on\_meet},\mbox{axiom})$

$k(n_0) = n_1\qquad\mbox{cnf}(\mbox{k\_on\_bottom},\mbox{axiom})$

$k(n_1) = n_0\qquad\mbox{cnf}(\mbox{k\_on\_top},\mbox{axiom})$

$k(k(\mbox{aa}))\vee ( \mbox{aa}\vee k(\mbox{aa}) ) = \mbox{aa}\vee k(\mbox{aa})\qquad\mbox{cnf}(\mbox{lhs}_1,\mbox{hypothesis})$

$k(k(k(\mbox{bb})))\vee ( \mbox{aa}\vee k(\mbox{aa}) ) = \mbox{aa}\vee k(\mbox{aa})\qquad\mbox{cnf}(\mbox{lhs}_2,\mbox{hypothesis})$

$k(k(\mbox{aa}))\vee ( k(\mbox{aa})\vee ( k(\mbox{bb})\vee k(\mbox{cc}) ) ) = k(\mbox{aa})\vee ( k(\mbox{bb})\vee k(\mbox{cc}) )\qquad\mbox{cnf}(\mbox{lhs}_3,\mbox{hypothesis})$

$k(k(k(\mbox{bb})))\vee ( k(\mbox{aa})\vee ( k(\mbox{bb})\vee k(\mbox{cc}) ) ) = k(\mbox{aa})\vee ( k(\mbox{bb})\vee k(\mbox{cc}) )\qquad\mbox{cnf}(\mbox{lhs}_4,\mbox{hypothesis})$

$k(k(\mbox{aa}\vee k(\mbox{bb})))\vee ( ( \mbox{aa}\wedge k(\mbox{bb}\wedge \mbox{cc}) )\vee k(\mbox{aa}) )\ne ( \mbox{aa}\wedge k(\mbox{bb}\wedge \mbox{cc}) )\vee k(\mbox{aa})\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT094-1.p} Axiom for weakly associative lattices (WAL), part 3






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\vee a\ne a\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT266-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT004-1.p} A fairly complex equation to establish

If X' = U $\vee$ V and Y' = U $\wedge$ V and U" = UC and V" = VC then
 (U $\vee$ V)" = (UC $\wedge$ VC). " means unique complement.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-3.ax'})$

$r_1'{=}a\vee b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_join\_b},\mbox{hypothesis})$

$r_2'{=}a\wedge b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_meet\_b},\mbox{hypothesis})$

$\mbox{unique\_complement}(a,a_2)\qquad\mbox{cnf}(\mbox{unique\_complement\_of\_a},\mbox{hypothesis})$

$\mbox{unique\_complement}(b,b_2)\qquad\mbox{cnf}(\mbox{unique\_complement\_of\_b},\mbox{hypothesis})$

$\neg\,\mbox{unique\_complement}(a\vee b,a_2\wedge b_2)\qquad\mbox{cnf}(\mbox{prove\_unique\_complement},\mbox{negated\_conjecture})$

\medskip\textbf{LAT103-1.p} Huntington equation H6 is independent of H10

Show that Huntington equation H10 does not imply Huntington
 equation H6 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{10},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT017-1.p} E2 holds in Ortholattices.

Prove that from ortholattice axioms, one can derive equation E2.




$\mbox{include}(\mbox{'Axioms/LAT003-0.ax'})$

$a\vee ( ( a'\wedge ( ( a\vee b' )\wedge ( a\vee b ) ) )\vee ( a'\wedge ( ( a'\wedge b )\vee ( a'\wedge b' ) ) ) )\ne n_1\qquad\mbox{cnf}(\mbox{prove\_e}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT003-1.p} A fairly complex equation to establish

If X' = U $\vee$ V and Y' = U $\wedge$ V and U" = UC and V" = VC then
 (U $\vee$ V)' = UC $\wedge$ VC. " means unique complement.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-3.ax'})$

$r_1'{=}a\vee b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_join\_b},\mbox{hypothesis})$

$r_2'{=}a\wedge b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_meet\_b},\mbox{hypothesis})$

$\mbox{unique\_complement}(a,a_2)\qquad\mbox{cnf}(\mbox{unique\_complement\_of\_a},\mbox{hypothesis})$

$\mbox{unique\_complement}(b,b_2)\qquad\mbox{cnf}(\mbox{unique\_complement\_of\_b},\mbox{hypothesis})$

$\neg\,a\vee b'{=}a_2\wedge b_2\qquad\mbox{cnf}(\mbox{prove\_complement},\mbox{negated\_conjecture})$

\medskip\textbf{LAT210-1.p} Equation H68 is Huntington by distributivity

Show that H68 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{68},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT209-1.p} Equation H64 is Huntington by distributivity

Show that H64 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge ( y\vee ( x\wedge z ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{64},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT038-1.p} Simplification rule in a distributive lattice

In a distributive lattice, the following simplification rule
 holds:
 forall a, b, c, d:
 if   f(a $\vee$ b, d) = f(c $\vee$ b, d) and
 f(a, d) \& f(b, d) = f(c, d) \& f(b, d)
 then f(a,d) = f(c,d).




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{dist\_join},\mbox{hypothesis})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{dist\_meet},\mbox{hypothesis})$

$f(u\vee v,w) = f(u,w)\vee f(v,w)\qquad\mbox{cnf}(\mbox{f\_on\_left\_join},\mbox{axiom})$

$f(n_0,w) = n_0\qquad\mbox{cnf}(\mbox{f\_on\_left\_bottom},\mbox{axiom})$

$f(w,u\vee v) = f(w,u)\vee f(w,v)\qquad\mbox{cnf}(\mbox{f\_on\_right\_join},\mbox{axiom})$

$f(w,n_0) = n_0\qquad\mbox{cnf}(\mbox{f\_on\_right\_bottom},\mbox{axiom})$

$f(\mbox{aa}\vee \mbox{bb},\mbox{dd}) = f(\mbox{cc}\vee \mbox{bb},\mbox{dd})\qquad\mbox{cnf}(\mbox{lhs}_1,\mbox{hypothesis})$

$f(\mbox{aa},\mbox{dd})\wedge f(\mbox{bb},\mbox{dd}) = f(\mbox{cc},\mbox{dd})\wedge f(\mbox{bb},\mbox{dd})\qquad\mbox{cnf}(\mbox{lhs}_2,\mbox{hypothesis})$

$f(\mbox{aa},\mbox{dd})\ne f(\mbox{cc},\mbox{dd})\qquad\mbox{cnf}(\mbox{rhs},\mbox{negated\_conjecture})$

\medskip\textbf{LAT389-1.p} Short axiom for lattice theory, part 1






$( ( a + b )\cdot ( b + a\cdot b ) + c )\cdot ( ( a + ( ( d + b )\cdot ( b + e ) )\cdot b )\cdot ( ( b + ( ( d\cdot ( b\cdot e ) + f\cdot b ) + b ) )\cdot ( v_6 + b\cdot ( ( b\cdot v_7 + f\cdot b ) + b ) ) + a\cdot ( ( ( d + b )\cdot ( b + e ) )\cdot b ) ) + ( ( a + b )\cdot ( b + a\cdot b ) )\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$a\cdot ( b + ( a + c ) )\ne a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT132-1.p} Huntington equation H42 is independent of H69\_dual

Show that Huntington equation H69\_dual does not imply Huntington
 equation H42 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = ( x\vee ( z\wedge ( x\vee y ) ) )\wedge ( x\vee ( y\wedge ( x\vee z ) ) )\qquad\mbox{cnf}(\mbox{equation\_H69\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT101-1.p} Huntington equation H10 is independent of H6

Show that Huntington equation H6 does not imply Huntington
 equation H10 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_6,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( a\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{LAT113-1.p} Huntington equation H40 is independent of H50

Show that Huntington equation H50 does not imply Huntington
 equation H40 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LAT072-1.p} Given single axiom OML-23A, prove associativity

Given a single axiom candidate OML-23A for orthomodular lattices
 (OML) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of associativity.




$f(f(f(f(b,a),f(a,c)),d),f(a,f(f(c,f(f(a,a),c)),c))) = a\qquad\mbox{cnf}(\mbox{oml\_23A},\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT273-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_S}\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_L},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT177-1.p} Huntington equation H79\_dual implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\vee ( ( x\vee ( y\wedge ( x\vee z ) ) )\wedge ( z\vee u ) )\qquad\mbox{cnf}(\mbox{equation\_H79\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT138-1.p} Huntington equation H7 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_7,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT096-1.p} Axiom for weakly associative lattices (WAL), part 5






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( ( a\vee b )\wedge ( c\vee b ) )\wedge b\ne b\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LAT083-1.p} Axiom for lattice theory, part 4






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\vee a\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LAT122-1.p} Huntington equation H55 is independent of H21\_dual

Show that Huntington equation H21\_dual does not imply Huntington
 equation H55 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( y\vee ( x\wedge ( y\vee z ) ) )\wedge ( z\vee ( x\wedge y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H21\_dual},\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne a\vee ( b\wedge ( c\vee ( a\wedge ( c\vee b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{55},\mbox{negated\_conjecture})$

\medskip\textbf{LAT121-1.p} Huntington equation H55 is independent of H18\_dual

Show that Huntington equation H18\_dual does not imply Huntington
 equation H55 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( x\vee y )\wedge ( ( x\vee z )\wedge ( y\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H18\_dual},\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne a\vee ( b\wedge ( c\vee ( a\wedge ( c\vee b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{55},\mbox{negated\_conjecture})$

\medskip\textbf{LAT125-1.p} Huntington equation H69 is independent of H34\_dual

Show that Huntington equation H34\_dual does not imply Huntington
 equation H69 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee u ) ) = x\vee ( y\wedge ( z\vee ( y\wedge ( u\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H34\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge ( c\vee ( a\wedge b ) ) )\vee ( a\wedge ( b\vee ( a\wedge c ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LAT035-1.p} Composition to form a join hemimorphism

In a lattice with 0,1, the composition of a unary join
 antihemimorphism and a lattice antimorphism is a join
 hemimorphism.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge n_0 = n_0\qquad\mbox{cnf}(\mbox{x\_meet}_0,\mbox{axiom})$

$x\vee n_0 = x\qquad\mbox{cnf}(\mbox{x\_join}_0,\mbox{axiom})$

$x\wedge n_1 = x\qquad\mbox{cnf}(\mbox{x\_meet}_1,\mbox{axiom})$

$x\vee n_1 = n_1\qquad\mbox{cnf}(\mbox{x\_join}_1,\mbox{axiom})$

$x\wedge z = x \ \Rightarrow \ z\wedge ( x\vee y ) = x\vee ( y\wedge z )\qquad\mbox{cnf}(\mbox{modular},\mbox{axiom})$

$k(u\vee v) = k(u)\wedge k(v)\qquad\mbox{cnf}(\mbox{k\_on\_join},\mbox{axiom})$

$k(u\wedge v) = k(u)\vee k(v)\qquad\mbox{cnf}(\mbox{k\_on\_meet},\mbox{axiom})$

$k(n_0) = n_1\qquad\mbox{cnf}(\mbox{k\_on\_bottom},\mbox{axiom})$

$k(n_1) = n_0\qquad\mbox{cnf}(\mbox{k\_on\_top},\mbox{axiom})$

$f(u\wedge v) = f(u)\vee f(v)\qquad\mbox{cnf}(\mbox{f\_on\_meet},\mbox{axiom})$

$f(n_1) = n_0\qquad\mbox{cnf}(\mbox{f\_on\_top},\mbox{axiom})$

$f(k(\mbox{aa}\vee \mbox{bb})) = f(k(\mbox{aa}))\vee f(k(\mbox{bb})) \ \Rightarrow \ f(k(n_0))\ne n_0\qquad\mbox{cnf}(\mbox{comp\_join\_hemimorphism},\mbox{negated\_conjecture})$

\medskip\textbf{LAT057-1.p} Lattice theory complement (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

\medskip\textbf{LAT168-1.p} Huntington equation H18\_dual implies H58






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( x\vee y )\wedge ( ( x\vee z )\wedge ( y\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H18\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{58},\mbox{negated\_conjecture})$

\medskip\textbf{LAT174-1.p} Huntington equation H76\_dual implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( y\wedge u ) ) ) = x\vee ( y\wedge ( z\vee ( u\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H76\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT194-1.p} Equation H32 is Huntington by distributivity

Show that H32 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\wedge u ) ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge u )\vee ( y\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{32},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT106-1.p} Huntington equation H3 is independent of H22

Show that Huntington equation H22 does not imply Huntington
 equation H3 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( z\vee ( x\wedge y ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{22},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( a\wedge ( c\vee ( a\wedge b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT120-1.p} Huntington equation H58 is independent of H10\_dual

Show that Huntington equation H10\_dual does not imply Huntington
 equation H58 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( y\vee z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H10\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{58},\mbox{negated\_conjecture})$

\medskip\textbf{LAT165-1.p} Huntington equation H76 implies H77






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{76},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( b\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( a\wedge ( b\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{77},\mbox{negated\_conjecture})$

\medskip\textbf{LAT264-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT098-1.p} Huntington equation H3 is independent of H2

Show that Huntington equation H2 does not imply Huntington
 equation H3 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge ( y\vee z ) )\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_2,\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( a\wedge ( c\vee ( a\wedge b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT005-3.p} SAM's lemma

Let L be a modular lattice with 0 and 1.  Suppose that A and
 B are elements of L such that (A $\vee$ B) and (A $\wedge$ B) both have
 not necessarily unique complements. Then, (A $\vee$ B)' =
 ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ B)) $\wedge$ ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ A)).




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$r_1'{=}a\vee b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_join\_b},\mbox{hypothesis})$

$r_2'{=}a\wedge b\qquad\mbox{cnf}(\mbox{complement\_of\_a\_meet\_b},\mbox{hypothesis})$

$r_1\ne ( r_1\vee ( r_2\wedge b ) )\wedge ( r_1\vee ( r_2\wedge a ) )\qquad\mbox{cnf}(\mbox{prove\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LAT203-1.p} Equation H57 is Huntington by distributivity

Show that H57 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( y\vee z ) ) ) = x\wedge ( y\vee ( ( x\vee y )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{57},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT169-1.p} Huntington equation H21\_dual implies H58






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( y\vee ( x\wedge ( y\vee z ) ) )\wedge ( z\vee ( x\wedge y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H21\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{58},\mbox{negated\_conjecture})$

\medskip\textbf{LAT243-1.p} Equation H57 is Huntington by implication

Show that H57 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( y\vee z ) ) ) = x\wedge ( y\vee ( ( x\vee y )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{57},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT272-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_S}\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT019-1.p} In quasilattices, a distributive law implies its dual.






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{distributivity\_law},\mbox{hypothesis})$

$a\vee ( b\wedge c )\ne ( a\vee b )\wedge ( a\vee c )\qquad\mbox{cnf}(\mbox{prove\_distributivity\_law\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT260-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

\medskip\textbf{LAT279-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\neg\,\mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LAT394-1.p} Ortholattices in terms of Sheffer stroke + usual operations: unit






$f(f(f(f(a,b),f(b,c)),d),f(b,f(f(b,f(f(a,a),a)),c))) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$\mbox{or}(a,b) = f(f(a,a),f(b,b))\qquad\mbox{cnf}(\mbox{sos}_{001},\mbox{axiom})$

$\mbox{and}(a,b) = f(f(a,b),f(a,b))\qquad\mbox{cnf}(\mbox{sos}_{002},\mbox{axiom})$

$\mbox{neg}(a) = f(a,a)\qquad\mbox{cnf}(\mbox{sos}_{003},\mbox{axiom})$

$f(x_0,f(x_0,x_0))\ne f(x_1,f(x_1,x_1))\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT161-1.p} Huntington equation H58 implies H59






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( ( x\vee y )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{58},\mbox{axiom})$

$a\wedge ( ( b\vee c )\wedge ( b\vee d ) )\ne a\wedge ( b\vee ( ( b\vee d )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{59},\mbox{negated\_conjecture})$

\medskip\textbf{LAT119-1.p} Huntington equation H3 is independent of H82

Show that Huntington equation H82 does not imply Huntington
 equation H3 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( ( y\wedge ( x\vee z ) )\vee ( z\wedge ( x\vee y ) ) ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{equation\_H}_{82},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( a\wedge ( c\vee ( a\wedge b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT268-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_x},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_Ix1\_M\_Ay1\_J\_A\_58\_Aorder\_A\_Idual\_Acl\_J\_A\_61\_61\_A\_Iy1\_M\_Ax1\_J\_A\_58\_Aorder\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT135-1.p} Huntington equation H39\_dual is independent of H68

Show that Huntington equation H68 does not imply Huntington
 equation H39\_dual in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{68},\mbox{axiom})$

$a\vee ( b\wedge ( c\vee ( a\wedge d ) ) )\ne a\vee ( b\wedge ( c\vee ( d\wedge ( a\vee c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H39\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT133-1.p} Huntington equation H6\_dual is independent of H55

Show that Huntington equation H55 does not imply Huntington
 equation H6\_dual in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$a\vee ( b\wedge ( a\vee c ) )\ne a\vee ( ( a\vee ( b\wedge ( a\vee c ) ) )\wedge ( c\vee ( a\wedge b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H6\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT144-1.p} Huntington equation H32 implies H2






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\wedge u ) ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge u )\vee ( y\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{32},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge ( b\vee c ) )\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT191-1.p} Equation H21 is Huntington by distributivity

Show that H21 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT067-1.p} Weak property 94-3 to make a uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( b\wedge ( a\vee c ) ) ) ) ) = a\wedge ( b\vee c )\qquad\mbox{cnf}(\mbox{c94}_3,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT244-1.p} Equation H58 is Huntington by implication

Show that H58 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee z ) = x\wedge ( y\vee ( ( x\vee y )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{58},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT205-1.p} Equation H59 is Huntington by distributivity

Show that H59 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee u )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{59},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT092-1.p} Axiom for weakly associative lattices (WAL), part 1






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) )\vee ( f\wedge ( b\vee ( ( ( b\vee d )\wedge ( e\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( b\wedge d )\vee ( e\wedge b ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\wedge a\ne a\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT220-1.p} Equation H3 is Huntington by implication

Show that H3 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( x\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_3,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT052-1.p} Modular lattice is not modular ortholattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$x\vee ( y\wedge ( x\vee z ) ) = ( x\vee y )\wedge ( x\vee z )\qquad\mbox{cnf}(\mbox{modular\_law},\mbox{axiom})$

$( a\vee b )'\ne a'\wedge b'\qquad\mbox{cnf}(\mbox{prove\_compatibility\_law},\mbox{negated\_conjecture})$

\medskip\textbf{LAT128-1.p} Huntington equation H3 is independent of H58\_dual

Show that Huntington equation H58\_dual does not imply Huntington
 equation H3 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge z ) = x\vee ( y\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H58\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( a\wedge ( c\vee ( a\wedge b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT382+1.p} Tarski-Knaster fixed point theorem 02, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isEmpty}_0(w_0)\iff \neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmpty},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtlseqdt}_0(w_0,w_1) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mLessRel},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mARefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{sdtlseqdt}_0(w_0,w_1)\mbox{ and }\mbox{sdtlseqdt}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1)\mbox{ and }\mbox{aElement}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{sdtlseqdt}_0(w_0,w_1)\mbox{ and }\mbox{sdtlseqdt}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mTrans},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aLowerBoundOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_1) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_2,w_3) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefLB},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aUpperBoundOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_1) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_3,w_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefUB},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aInfimumOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\mbox{aLowerBoundOfIn}_0(w_2,w_1,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aLowerBoundOfIn}_0(w_3,w_1,w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_3,w_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefInf},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2{:}\;( \mbox{aSupremumOfIn}_0(w_2,w_1,w_0)\iff ( \mbox{aElementOf}_0(w_2,w_0)\mbox{ and }\mbox{aUpperBoundOfIn}_0(w_2,w_1,w_0)\mbox{ and }\forall w_3{:}\;( \mbox{aUpperBoundOfIn}_0(w_3,w_1,w_0) \ \Rightarrow \ \mbox{sdtlseqdt}_0(w_2,w_3) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSup},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \forall w_2,w_3{:}\;( ( \mbox{aSupremumOfIn}_0(w_2,w_1,w_0)\mbox{ and }\mbox{aSupremumOfIn}_0(w_3,w_1,w_0) ) \ \Rightarrow \ w_2 = w_3 ) ) )\qquad\mbox{fof}(\mbox{mSupUn},\mbox{axiom})$

$\mbox{aSet}_0(\mbox{xT})\qquad\mbox{fof}(\mbox{m\_}_{773},\mbox{hypothesis})$

$\mbox{aSubsetOf}_0(\mbox{xS},\mbox{xT})\qquad\mbox{fof}(\mbox{m\_\_773}_{01},\mbox{hypothesis})$

$\mbox{aInfimumOfIn}_0(\mbox{xu},\mbox{xS},\mbox{xT})\mbox{ and }\mbox{aInfimumOfIn}_0(\mbox{xv},\mbox{xS},\mbox{xT})\qquad\mbox{fof}(\mbox{m\_}_{792},\mbox{hypothesis})$

$\mbox{xu} = \mbox{xv}\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{LAT173-1.p} Huntington equation H76\_dual implies H40






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( y\wedge u ) ) ) = x\vee ( y\wedge ( z\vee ( u\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H76\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LAT213-1.p} Equation H76 is Huntington by distributivity

Show that H76 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{76},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT270-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_S}\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT054-1.p} Countermodel for Megill equation for ortholattices






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )' = x'\wedge y'\qquad\mbox{cnf}(\mbox{compatibility}_1,\mbox{axiom})$

$( x\wedge y )' = x'\vee y'\qquad\mbox{cnf}(\mbox{compatibility}_2,\mbox{axiom})$

$x'\vee x = n_1\qquad\mbox{cnf}(\mbox{invertability}_1,\mbox{axiom})$

$x'\wedge x = n_0\qquad\mbox{cnf}(\mbox{invertability}_2,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{invertability}_3,\mbox{axiom})$

$a\vee ( b'\wedge ( a'\vee ( b'\wedge ( a\vee ( b'\wedge a' ) ) ) ) )\ne a\vee ( b'\wedge ( a'\vee ( b'\wedge ( a\vee ( b'\wedge ( a'\vee ( b'\wedge a ) ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LAT155-1.p} Huntington equation H49 implies H2






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( ( x\wedge z )\vee ( z\wedge ( y\vee u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{49},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge ( b\vee c ) )\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT151-1.p} Huntington equation H39 implies H42






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{39},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT245-1.p} Equation H59 is Huntington by implication

Show that H59 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee u )\wedge ( z\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{59},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT015-1.p} Single axiom for lattice theory

This starts with a single axiom for lattice theory and derives a
 standard basis for lattice theory.




$( ( ( x\wedge y )\vee ( y\wedge ( x\vee y ) ) )\wedge z )\vee ( ( ( x\wedge ( ( ( x_1\wedge y )\vee ( y\wedge x_2 ) )\vee y ) )\vee ( ( ( y\wedge ( ( ( x_1\vee ( y\vee x_2 ) )\wedge ( x_3\vee y ) )\wedge y ) )\vee ( u\wedge ( y\vee ( ( ( x_1\vee ( y\vee x_2 ) )\wedge ( x_3\vee y ) )\wedge y ) ) ) )\wedge ( x\vee ( ( ( x_1\wedge y )\vee ( y\wedge x_2 ) )\vee y ) ) ) )\wedge ( ( ( x\wedge y )\vee ( y\wedge ( x\vee y ) ) )\vee z ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a\wedge a = a\mbox{ and }a\wedge b = b\wedge a\mbox{ and }( a\wedge b )\wedge c = a\wedge ( b\wedge c )\mbox{ and }a\vee a = a\mbox{ and }a\vee b = b\vee a\mbox{ and }( a\vee b )\vee c = a\vee ( b\vee c )\mbox{ and }a\wedge ( a\vee b ) = a ) \ \Rightarrow \ a\vee ( a\wedge b )\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{LAT134-1.p} Huntington equation H22\_dual is independent of H61

Show that Huntington equation H61 does not imply Huntington
 equation H22\_dual in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( x\vee y )\wedge ( ( x\wedge y )\vee z ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{61},\mbox{axiom})$

$( a\vee b )\wedge ( a\vee c )\ne a\vee ( ( b\vee ( c\wedge ( a\vee b ) ) )\wedge ( c\vee ( a\wedge b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H22\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT068-1.p} Weak property F53 to make a uniquely complemented lattice Boolean






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee a' = n_1\qquad\mbox{cnf}(\mbox{top},\mbox{axiom})$

$a\wedge a' = n_0\qquad\mbox{cnf}(\mbox{bottom},\mbox{axiom})$

$( a\vee b = n_1\mbox{ and }a\wedge b = n_0 ) \ \Rightarrow \ a' = b\qquad\mbox{cnf}(\mbox{complements\_are\_unique},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( d\vee ( a\wedge ( b\vee c ) ) ) ) ) = a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\qquad\mbox{cnf}(f_{53},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT079-1.p} Given single axiom MOL-27B2, prove modularity

Given a single axiom candidate MOL-27B2 for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of modularity.




$f(f(f(f(b,a),f(a,c)),d),f(a,f(f(f(b,f(b,f(f(c,c),a))),a),c))) = a\qquad\mbox{cnf}(\mbox{mol\_27B}_2,\mbox{axiom})$

$f(a,f(b,f(a,f(c,c))))\ne f(a,f(c,f(a,f(b,b))))\qquad\mbox{cnf}(\mbox{modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT257-1.p} Equation H82 is Huntington by implication

Show that H82 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\wedge ( x\vee z ) )\vee ( z\wedge ( x\vee y ) ) ) = ( x\wedge y )\vee ( x\wedge z )\qquad\mbox{cnf}(\mbox{equation\_H}_{82},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT391-1.p} Short axiom for lattice theory, part 3






$( ( a + b )\cdot ( b + a\cdot b ) + c )\cdot ( ( a + ( ( d + b )\cdot ( b + e ) )\cdot b )\cdot ( ( b + ( ( d\cdot ( b\cdot e ) + f\cdot b ) + b ) )\cdot ( v_6 + b\cdot ( ( b\cdot v_7 + f\cdot b ) + b ) ) + a\cdot ( ( ( d + b )\cdot ( b + e ) )\cdot b ) ) + ( ( a + b )\cdot ( b + a\cdot b ) )\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$( ( a + b )\cdot ( b + c ) )\cdot b\ne b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT237-1.p} Equation H40 is Huntington by implication

Show that H40 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{40},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT142-1.p} Huntington equation H22 implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( z\vee ( x\wedge y ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{22},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT182-1.p} Equation H6 is Huntington by distributivity

Show that H6 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_6,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT112-1.p} Huntington equation H42 is independent of H47

Show that Huntington equation H47 does not imply Huntington
 equation H42 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\wedge ( z\vee ( y\wedge u ) ) ) = x\wedge ( y\wedge ( z\vee ( u\wedge ( y\vee ( x\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{47},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT184-1.p} Equation H8 is Huntington by distributivity

Show that H8 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = ( x\wedge y )\vee ( x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_8,\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT206-1.p} Equation H60 is Huntington by distributivity

Show that H60 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee z )\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{60},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT034-1.p} Idempotency of meet






$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption}_1,\mbox{axiom})$

$x\vee ( x\wedge y ) = x\qquad\mbox{cnf}(\mbox{absorption}_2,\mbox{axiom})$

$x\wedge y = y\wedge x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet},\mbox{axiom})$

$x\vee y = y\vee x\qquad\mbox{cnf}(\mbox{commutativity\_of\_join},\mbox{axiom})$

$( x\wedge y )\wedge z = x\wedge ( y\wedge z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet},\mbox{axiom})$

$( x\vee y )\vee z = x\vee ( y\vee z )\qquad\mbox{cnf}(\mbox{associativity\_of\_join},\mbox{axiom})$

$\mbox{xx}\wedge \mbox{xx}\ne \mbox{xx}\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet},\mbox{negated\_conjecture})$

\medskip\textbf{LAT227-1.p} Equation H15 is Huntington by implication

Show that H15 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge y )\vee ( ( x\wedge z )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{15},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT171-1.p} Huntington equation H61\_dual implies H6






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( x\wedge y )\vee ( ( x\vee y )\wedge z ) )\qquad\mbox{cnf}(\mbox{equation\_H61\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( ( a\wedge ( b\vee ( a\wedge c ) ) )\vee ( c\wedge ( a\vee b ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LAT070-1.p} Given single axiom OL-23A, prove associativity

Given a single axiom OL-23A for ortholattices (OL) in terms of
 the Sheffer Stroke, prove a Sheffer stroke form of associativity.




$f(f(f(f(b,a),f(a,c)),d),f(a,f(f(a,f(f(b,b),b)),c))) = a\qquad\mbox{cnf}(\mbox{ol\_23A},\mbox{axiom})$

$f(a,f(f(b,c),f(b,c)))\ne f(c,f(f(b,a),f(b,a)))\qquad\mbox{cnf}(\mbox{associativity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT269-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{v\_intY}_1,\mbox{c\_Tarski\_Oinduced}(\mbox{v\_intY}_1,\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Product\_\_Type\_OUnity},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OTop\_Acl\_A\_58\_AA\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Tarski\_Opotype\_Opotype\_\_ext}(\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{c\_Tarski\_Oinduced}(\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Product\_\_Type\_OUnity},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ or }\mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_emptyset} )\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_Aa2\_A\_58\_AA\_59\_Ab2\_A\_58\_AA\_59\_Ainterval\_Ar\_Aa2\_Ab2\_A\_126\_61\_A\_123\_125\_A\_124\_93\_A\_61\_61\_62\_A\_I\_124\_Apset\_A\_61\_Ainterval\_Ar\_Aa2\_Ab2\_M\_Aorder\_A\_61\_Ainduced\_A\_Iinterval\_Ar\_Aa2\_Ab2\_J\_Ar\_A\_124\_J\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{v\_intY}_1 = \mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{c\_Tarski\_Olub}(\mbox{v\_Y},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OintY1\_A\_61\_61\_Ainterval\_Ar\_A\_Ilub\_AY\_Acl\_J\_A\_ITop\_Acl\_J}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_Y},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Olub\_AY\_Acl\_A\_58\_AA\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_Tarski\_Ointerval}(\mbox{v\_r},\mbox{v\_x},\mbox{c\_Tarski\_OTop}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\ne \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ox1\_A\_58\_AA\_A\_61\_61\_62\_Ainterval\_Ar\_Ax1\_A\_ITop\_Acl\_J\_A\_61\_A\_123\_125\_A\_61\_61\_AFalse}_0,\mbox{axiom})$

\medskip\textbf{LAT256-1.p} Equation H81 is Huntington by implication

Show that H81 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee ( y\wedge ( x\vee z ) ) ) ) ) = x\wedge ( ( x\wedge y )\vee ( x\wedge z ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{81},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT393-1.p} Ortholattices in terms of Sheffer stroke + ops: associativity






$f(f(f(f(a,b),f(b,c)),d),f(b,f(f(b,f(f(a,a),a)),c))) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$\mbox{or}(a,b) = f(f(a,a),f(b,b))\qquad\mbox{cnf}(\mbox{sos}_{001},\mbox{axiom})$

$\mbox{and}(a,b) = f(f(a,b),f(a,b))\qquad\mbox{cnf}(\mbox{sos}_{002},\mbox{axiom})$

$\mbox{neg}(a) = f(a,a)\qquad\mbox{cnf}(\mbox{sos}_{003},\mbox{axiom})$

$f(x_0,f(f(x_1,x_2),f(x_1,x_2)))\ne f(x_1,f(f(x_0,x_2),f(x_0,x_2)))\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{LAT253-1.p} Equation H76 is Huntington by implication

Show that H76 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{76},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT153-1.p} Huntington equation H40 implies H7






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{40},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( a\wedge ( ( a\wedge b )\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT260-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_in\_\_lattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT011-1.p} Uniqueness of meet (dually join) in lattice theory

Let's say we have a lattice with two meet operations, say
 meet1 and meet2.  In other words, {join,meet1} is a lattice,
 and {join,meet2} is a lattice.  Then, we can prove that the
 two meet operations are really the same.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge_2 x = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_meet}_2,\mbox{axiom})$

$x\wedge_2 y = y\wedge_2 x\qquad\mbox{cnf}(\mbox{commutativity\_of\_meet}_2,\mbox{axiom})$

$x\wedge_2 ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption1}_2,\mbox{axiom})$

$x\vee ( x\wedge_2 y ) = x\qquad\mbox{cnf}(\mbox{absorption2}_2,\mbox{axiom})$

$( x\wedge_2 y )\wedge_2 z = x\wedge_2 ( y\wedge_2 z )\qquad\mbox{cnf}(\mbox{associativity\_of\_meet}_2,\mbox{axiom})$

$a\wedge b\ne a\wedge_2 b\qquad\mbox{cnf}(\mbox{prove\_meets\_are\_same},\mbox{negated\_conjecture})$

\medskip\textbf{LAT105-1.p} Huntington equation H10 is independent of H21

Show that Huntington equation H21 does not imply Huntington
 equation H10 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( c\wedge ( a\vee ( b\wedge c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{LAT008-1.p} Sholander's basis for distributive lattices, part 5 (of 6).

This is part of the proof that Sholanders 2-basis for
 distributive lattices is correct. Here we prove the absorption
 law  x $\vee$ (x $\wedge$ y) = x.




$x\wedge ( x\vee y ) = x\qquad\mbox{cnf}(\mbox{absorption},\mbox{axiom})$

$x\wedge ( y\vee z ) = ( z\wedge x )\vee ( y\wedge x )\qquad\mbox{cnf}(\mbox{distribution},\mbox{axiom})$

$a\vee ( a\wedge b )\ne a\qquad\mbox{cnf}(\mbox{prove\_absorbtion\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT281-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Otrans\_\_def}_0,\mbox{axiom})$

$\mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Main\_Otrans\_\_def\_}_1(\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Main\_Otrans\_\_def\_}_2(\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Otrans\_\_def}_1,\mbox{axiom})$

$\mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Main\_Otrans\_\_def\_}_2(\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Main\_Otrans\_\_def\_}_3(\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_Otrans\_\_def}_2,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Main\_Otrans\_\_def\_}_1(\mbox{v\_r},\mbox{t\_a}),\mbox{c\_Main\_Otrans\_\_def\_}_3(\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Relation\_Otrans\_\_def}_3,\mbox{axiom})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_2,\mbox{axiom})$

$( \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ and }\mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT077-1.p} Given single axiom MOL-27B1, prove modularity

Given a single axiom candidate MOL-27B1 for modular ortholattices
 (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form
 of modularity.




$f(f(f(f(b,a),f(c,a)),d),f(a,f(f(f(f(f(f(b,b),a),c),c),a),b))) = a\qquad\mbox{cnf}(\mbox{mol\_27B}_1,\mbox{axiom})$

$f(a,f(b,f(a,f(c,c))))\ne f(a,f(c,f(a,f(b,b))))\qquad\mbox{cnf}(\mbox{modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT193-1.p} Equation H24 is Huntington by distributivity

Show that H24 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( y\wedge z ) = ( x\wedge y )\vee ( y\wedge ( ( x\wedge y )\vee ( z\wedge ( x\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{24},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT279-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_Relation\_Oantisym}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_P},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_P},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OPartialOrder\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT258+1.p} A duality result on distributivity in lattices






$x\vee y{=}t\mbox{ and }x\vee z{=}u\qquad\mbox{fof}(\mbox{join\_assumption},\mbox{axiom})$

$t\wedge u{=}v\qquad\mbox{fof}(\mbox{meet\_assumption},\mbox{axiom})$

$y\wedge z{=}w\mbox{ and }x\vee w{=}p\qquad\mbox{fof}(\mbox{meet\_join\_assumption},\mbox{axiom})$

$v < p \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goal\_ax},\mbox{axiom})$

$\forall a{:}\;a < a\qquad\mbox{fof}(\mbox{less\_than\_reflexive},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a < b\mbox{ and }b < c ) \ \Rightarrow \ a < c )\qquad\mbox{fof}(\mbox{less\_than\_transitive},\mbox{axiom})$

$\forall a,b,c{:}\;( a\wedge b{=}c \ \Rightarrow \ ( c < a\mbox{ and }c < b ) )\qquad\mbox{fof}(\mbox{lower\_bound\_meet},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\wedge b{=}c\mbox{ and }d < a\mbox{ and }d < b ) \ \Rightarrow \ d < c )\qquad\mbox{fof}(\mbox{greatest\_lower\_bound\_meet},\mbox{axiom})$

$\forall a,b,c{:}\;( a\vee b{=}c \ \Rightarrow \ ( a < c\mbox{ and }b < c ) )\qquad\mbox{fof}(\mbox{upper\_bound\_join},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\vee b{=}c\mbox{ and }a < d\mbox{ and }b < d ) \ \Rightarrow \ c < d )\qquad\mbox{fof}(\mbox{least\_upper\_bound\_join},\mbox{axiom})$

$\forall a,b{:}\;( a < b \ \Rightarrow \ ( a\wedge b{=}a\mbox{ and }a\vee b{=}b ) )\qquad\mbox{fof}(\mbox{less\_than\_meet\_join},\mbox{axiom})$

$\forall a,b,c{:}\;( a\wedge b{=}c \ \Rightarrow \ b\wedge a{=}c )\qquad\mbox{fof}(\mbox{commutitivity\_meet},\mbox{axiom})$

$\forall a,b,c{:}\;( a\vee b{=}c \ \Rightarrow \ b\vee a{=}c )\qquad\mbox{fof}(\mbox{commutitivity\_join},\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( ( a\wedge b{=}d\mbox{ and }d\wedge c{=}e\mbox{ and }b\wedge c{=}f ) \ \Rightarrow \ a\wedge f{=}e )\qquad\mbox{fof}(\mbox{associativity\_meet},\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( ( a\vee b{=}d\mbox{ and }d\vee c{=}e\mbox{ and }b\vee c{=}f ) \ \Rightarrow \ a\vee f{=}e )\qquad\mbox{fof}(\mbox{associativity\_join},\mbox{axiom})$

$\forall a,b,c,d,e,f,g,h{:}\;( ( b\vee c{=}h\mbox{ and }a\wedge h{=}d\mbox{ and }a\wedge b{=}e\mbox{ and }a\wedge c{=}f\mbox{ and }e\vee f{=}g ) \ \Rightarrow \ d < g )\qquad\mbox{fof}(\mbox{lo\_le\_distr},\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;a\wedge b{=}c\qquad\mbox{fof}(\mbox{do\_lattice},\mbox{axiom})$

$\mbox{goal}\qquad\mbox{fof}(\mbox{goal\_to\_be\_proved},\mbox{conjecture})$

\medskip\textbf{LAT228-1.p} Equation H16 is Huntington by implication

Show that H16 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge y )\vee ( z\wedge ( y\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{16},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT148-1.p} Huntington equation H34 implies H7






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\wedge ( y\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{34},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge c ) )\ne a\wedge ( b\vee ( a\wedge ( ( a\wedge b )\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_7,\mbox{negated\_conjecture})$

\medskip\textbf{LAT254-1.p} Equation H79 is Huntington by implication

Show that H79 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge u ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{79},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT276-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_V},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xa},\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_xa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_{10},\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_xa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_xa},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_xa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xa},\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_S},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_9,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_L},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{v\_sko\_\_4mP}(\mbox{v\_L},\mbox{v\_S},\mbox{v\_r}),\mbox{v\_S},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AS1\_A\_60\_61\_AA\_59\_AL1\_A\_58\_AA\_59\_AALL\_Ax\_58S1\_O\_A\_Ix\_M\_AL1\_J\_A\_58\_Ar\_A\_124\_93\_A\_61\_61\_62\_A\_Ilub\_AS1\_Acl\_M\_AL1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_L},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_sko\_\_4mP}(\mbox{v\_L},\mbox{v\_S},\mbox{v\_r}),\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AS1\_A\_60\_61\_AA\_59\_AL1\_A\_58\_AA\_59\_AALL\_Ax\_58S1\_O\_A\_Ix\_M\_AL1\_J\_A\_58\_Ar\_A\_124\_93\_A\_61\_61\_62\_A\_Ilub\_AS1\_Acl\_M\_AL1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AS1\_A\_60\_61\_AA\_59\_Ax1\_A\_58\_AS1\_A\_124\_93\_A\_61\_61\_62\_A\_Ix1\_M\_Alub\_AS1\_Acl\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{v\_r} = \mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Or\_A\_61\_61\_Aorder\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT005-4.p} SAM's lemma

Let L be a modular lattice with 0 and 1.  Suppose that A and
 B are elements of L such that (A $\vee$ B) and (A $\wedge$ B) both have
 not necessarily unique complements. Then, (A $\vee$ B)' =
 ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ B)) $\wedge$ ((A $\vee$ B)' $\vee$ ((A $\wedge$ B)' $\wedge$ A)).




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$r_2\vee ( a\wedge b ) = n_1\qquad\mbox{cnf}(\mbox{r2\_complement\_meet\_a\_b}_1,\mbox{negated\_conjecture})$

$r_2\wedge ( a\wedge b ) = n_0\qquad\mbox{cnf}(\mbox{r2\_complement\_meet\_a\_b}_2,\mbox{negated\_conjecture})$

$r_1\vee ( a\vee b ) = n_1\qquad\mbox{cnf}(\mbox{r1\_complement\_join\_a\_b}_1,\mbox{negated\_conjecture})$

$r_1\wedge ( a\vee b ) = n_0\qquad\mbox{cnf}(\mbox{r1\_complement\_join\_a\_b}_2,\mbox{negated\_conjecture})$

$r_1\vee ( a\wedge r_2 ) = b_2\qquad\mbox{cnf}(\mbox{define\_b}_2,\mbox{negated\_conjecture})$

$r_1\vee ( b\wedge r_2 ) = a_2\qquad\mbox{cnf}(\mbox{define\_a}_2,\mbox{negated\_conjecture})$

$a_2\wedge b_2\ne r_1\qquad\mbox{cnf}(\mbox{prove\_SAMs\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LAT231-1.p} Equation H21 is Huntington by implication

Show that H21 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\wedge y )\vee ( x\wedge z ) = x\wedge ( ( y\wedge ( x\vee ( y\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{21},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT163-1.p} Huntington equation H76 implies H32






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( x\wedge y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{76},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\wedge d ) ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge d )\vee ( b\wedge d ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{LAT207-1.p} Equation H61 is Huntington by distributivity

Show that H61 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$( x\vee y )\wedge ( x\vee z ) = x\vee ( ( x\vee y )\wedge ( ( x\wedge y )\vee z ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{61},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT219-1.p} Equation H2 is Huntington by implication

Show that H2 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge ( y\vee z ) )\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_2,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT396-1.p} Lattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

\medskip\textbf{LAT395-1.p} Lattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-3.ax'})$

\medskip\textbf{LAT200-1.p} Equation H50 is Huntington by distributivity

Show that H50 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge ( y\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{50},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT263-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_in\_\_lattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Oantisym}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Aantisym\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Orefl}(\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Arefl\_A\_Ipset\_Acl1\_J\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})) \ \Rightarrow \ \mbox{c\_Relation\_Otrans}(\mbox{c\_Tarski\_Opotype\_Oorder}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl1\_A\_58\_ACompleteLattice\_A\_61\_61\_62\_Atrans\_A\_Iorder\_Acl1\_J\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Ocl\_A\_58\_ACompleteLattice\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT047-1.p} Lattice is not modular lattice






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$a\vee ( b\wedge ( a\vee c ) )\ne ( a\vee b )\wedge ( a\vee c )\qquad\mbox{cnf}(\mbox{prove\_modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT188-1.p} Equation H16 is Huntington by distributivity

Show that H16 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( x\wedge y )\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge y )\vee ( z\wedge ( y\vee ( z\wedge ( x\vee y ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{16},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT267-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT136-1.p} Huntington equation H39\_dual is independent of H69

Show that Huntington equation H69 does not imply Huntington
 equation H39\_dual in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee z ) = ( x\wedge ( z\vee ( x\wedge y ) ) )\vee ( x\wedge ( y\vee ( x\wedge z ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{69},\mbox{axiom})$

$a\vee ( b\wedge ( c\vee ( a\wedge d ) ) )\ne a\vee ( b\wedge ( c\vee ( d\wedge ( a\vee c ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H39\_dual},\mbox{negated\_conjecture})$

\medskip\textbf{LAT263-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_in}(\mbox{v\_cl},\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OCL\_Olub\_\_in\_\_lattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OPartialOrder},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_APartialOrder}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Oglb}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}) = \mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Oglb\_\_dual\_\_lub}_0,\mbox{axiom})$

$\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}) = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Opset\_A\_Idual\_Acl\_J\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

\medskip\textbf{LAT195-1.p} Equation H34 is Huntington by distributivity

Show that H34 is Huntington by deriving distributivity in uniquely
 complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\wedge ( y\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{34},\mbox{axiom})$

$a\wedge ( b\vee c )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT248-1.p} Equation H63 is Huntington by implication

Show that H63 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( ( y\vee z )\wedge ( y\vee u ) ) = x\wedge ( y\vee ( ( y\vee z )\wedge ( u\vee ( y\wedge z ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{63},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT218-1.p} Equation H1 is Huntington by implication

Show that H1 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge ( x\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( z\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_1,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT023-1.p} Yet another modularity equation for quasilattices.






$\mbox{include}(\mbox{'Axioms/LAT004-0.ax'})$

$( ( x\vee y )\wedge z )\vee y = ( ( z\vee y )\wedge x )\vee y\qquad\mbox{cnf}(\mbox{modularity\_axiom},\mbox{hypothesis})$

$a\wedge ( b\vee ( a\wedge c ) )\ne ( a\wedge b )\vee ( a\wedge c )\qquad\mbox{cnf}(\mbox{prove\_modularity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT167-1.p} Huntington equation H78 implies H77






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\wedge ( y\vee u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\vee ( y\wedge ( x\vee u ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{78},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( b\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( a\wedge ( b\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{77},\mbox{negated\_conjecture})$

\medskip\textbf{LAT029-1.p} Absorption basis for WAL

Prove that the 5 absorption equations below are a basis for
 weakly associative lattices.  This can be done by deriving
 commutativity and idempotence of the two operations.




$( x\wedge y )\vee ( x\wedge ( x\vee y ) ) = x\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_1,\mbox{axiom})$

$( x\wedge x )\vee ( y\wedge ( x\vee x ) ) = x\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_2,\mbox{axiom})$

$( x\wedge y )\vee ( y\wedge ( x\vee y ) ) = y\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_3,\mbox{axiom})$

$( ( x\vee y )\wedge ( z\vee x ) )\wedge x = x\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_4,\mbox{axiom})$

$( ( x\wedge y )\vee ( z\wedge x ) )\vee x = x\qquad\mbox{cnf}(\mbox{wal\_absorbtion}_5,\mbox{axiom})$

$( a\wedge a = a\mbox{ and }b\wedge a = a\wedge b\mbox{ and }a\vee a = a ) \ \Rightarrow \ b\vee a\ne a\vee b\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{LAT272-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

$( \mbox{c\_Tarski\_OisLub}(\mbox{v\_S},\mbox{v\_cl},\mbox{v\_L},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_y},\mbox{v\_S},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_L},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_O\_91\_124\_AisLub\_AS1\_Acl\_AL1\_59\_Ay1\_A\_58\_AS1\_A\_124\_93\_A\_61\_61\_62\_A\_Iy1\_M\_AL1\_J\_A\_58\_Ar\_A\_61\_61\_ATrue}_0,\mbox{axiom})$

\medskip\textbf{LAT009-1.p} A self-dual form of distributivity for lattice theory.

From lattice theory axioms and a self-dual form of
 distributivity, we prove ordinary distributivity.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$( ( ( x\wedge y )\vee z )\wedge y )\vee ( z\wedge x ) = ( ( ( x\vee y )\wedge z )\vee y )\wedge ( z\vee x )\qquad\mbox{cnf}(\mbox{distributivity\_dual},\mbox{axiom})$

$a\vee ( b\wedge c )\ne ( a\vee b )\wedge ( a\vee c )\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT234-1.p} Equation H32 is Huntington by implication

Show that H32 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\wedge u ) ) ) = x\wedge ( y\vee ( z\wedge ( ( x\wedge u )\vee ( y\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{32},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT170-1.p} Huntington equation H49\_dual implies H58






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\vee ( y\wedge ( ( x\vee z )\wedge ( z\vee ( y\wedge u ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H49\_dual},\mbox{axiom})$

$a\wedge ( b\vee c )\ne a\wedge ( b\vee ( ( a\vee b )\wedge ( c\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{58},\mbox{negated\_conjecture})$

\medskip\textbf{LAT116-1.p} Huntington equation H60 is independent of H55

Show that Huntington equation H55 does not imply Huntington
 equation H60 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( x\vee z ) ) = x\vee ( y\wedge ( z\vee ( x\wedge ( z\vee y ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{55},\mbox{axiom})$

$a\wedge ( ( b\vee c )\wedge ( b\vee d ) )\ne a\wedge ( b\vee ( ( b\vee c )\wedge ( d\vee ( a\wedge b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{60},\mbox{negated\_conjecture})$

\medskip\textbf{LAT109-1.p} Huntington equation H40 is independent of H37

Show that Huntington equation H37 does not imply Huntington
 equation H40 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( z\vee ( x\wedge u ) ) ) = x\wedge ( y\vee ( z\vee ( u\wedge ( x\vee ( y\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{37},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( d\vee ( c\wedge ( a\vee b ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LAT265-2.p} Problem about Tarski's fixed point theorem






$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_Tarski\_Odual\_Acl\_A\_58\_ACompleteLattice}_0,\mbox{axiom})$

\medskip\textbf{LAT222-1.p} Equation H6 is Huntington by implication

Show that H6 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( ( x\wedge ( y\vee ( x\wedge z ) ) )\vee ( z\wedge ( x\vee y ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_6,\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT226-1.p} Equation H11 is Huntington by implication

Show that H11 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( x\wedge z ) ) = x\wedge ( y\vee ( z\wedge ( x\vee ( y\wedge ( z\vee ( x\wedge y ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{11},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT030-1.p} Single axiom for weakly associative lattices (WAL)

This starts with a single axiom for WAL and derives a
 standard basis for WAL.




$( ( ( x\wedge y )\vee ( y\wedge ( x\vee y ) ) )\wedge z )\vee ( ( ( x\wedge ( ( ( y\wedge x_1 )\vee ( x_2\wedge y ) )\vee y ) )\vee ( ( ( y\wedge ( ( ( y\vee x_1 )\wedge ( x_2\vee y ) )\wedge y ) )\vee ( u\wedge ( y\vee ( ( ( y\vee x_1 )\wedge ( x_2\vee y ) )\wedge y ) ) ) )\wedge ( x\vee ( ( ( y\wedge x_1 )\vee ( x_2\wedge y ) )\vee y ) ) ) )\wedge ( ( ( x\wedge y )\vee ( y\wedge ( x\vee y ) ) )\vee z ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a\wedge a = a\mbox{ and }b\wedge a = a\wedge b\mbox{ and }a\vee a = a\mbox{ and }b\vee a = a\vee b\mbox{ and }( ( a\vee b )\wedge ( c\vee b ) )\wedge b = b ) \ \Rightarrow \ ( ( a\wedge b )\vee ( c\wedge b ) )\vee b\ne b\qquad\mbox{cnf}(\mbox{prove\_wal\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{LAT108-1.p} Huntington equation H42 is independent of H31

Show that Huntington equation H31 does not imply Huntington
 equation H42 in lattice theory.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\wedge ( y\vee ( x\wedge ( z\wedge u ) ) ) = x\wedge ( y\vee ( z\wedge ( u\wedge ( y\vee ( x\wedge z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{31},\mbox{axiom})$

$a\wedge ( b\vee ( c\wedge ( a\vee d ) ) )\ne a\wedge ( b\vee ( c\wedge ( b\vee ( d\vee ( a\wedge c ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LAT275-2.p} Problem about Tarski's fixed point theorem






$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_A} = \mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OA\_A\_61\_61\_Apset\_Acl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Tarski\_Olub}(\mbox{v\_S},\mbox{v\_cl},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Tarski\_OS\_A\_60\_61\_AA\_A\_61\_61\_62\_Alub\_AS\_Acl\_A\_58\_AA}_0,\mbox{axiom})$

\medskip\textbf{LAT235-1.p} Equation H34 is Huntington by implication

Show that H34 is Huntington by deriving the Huntington implication
 X $\wedge$ Y = Y  $\to$  X' $\vee$ Y' = Y' in uniquely complemented lattices.




$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-4.ax'})$

$x\wedge ( y\vee ( z\wedge u ) ) = x\wedge ( y\vee ( z\wedge ( y\vee ( u\wedge ( y\vee z ) ) ) ) )\qquad\mbox{cnf}(\mbox{equation\_H}_{34},\mbox{axiom})$

$b\wedge a = a\qquad\mbox{cnf}(\mbox{prove\_distributivity\_hypothesis},\mbox{hypothesis})$

$b'\vee a'\ne a'\qquad\mbox{cnf}(\mbox{prove\_distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{LAT265-1.p} Problem about Tarski's fixed point theorem






$\mbox{include}(\mbox{'Axioms/LAT006-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT006-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_Tarski\_Opotype\_Opset}(\mbox{v\_cl},\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Tarski\_Odual}(\mbox{v\_cl},\mbox{t\_a}),\mbox{c\_Tarski\_OCompleteLattice},\mbox{tc\_Tarski\_Opotype\_Opotype\_\_ext\_\_type}(\mbox{t\_a},\mbox{tc\_Product\_\_Type\_Ounit}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LAT080-1.p} Axiom for lattice theory, part 1






$( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\wedge c )\vee ( ( ( a\wedge ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) )\vee ( ( ( b\wedge ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) )\vee ( g\wedge ( b\vee ( ( ( d\vee ( b\vee e ) )\wedge ( f\vee b ) )\wedge b ) ) ) )\wedge ( a\vee ( ( ( d\wedge b )\vee ( b\wedge e ) )\vee b ) ) ) )\wedge ( ( ( a\wedge b )\vee ( b\wedge ( a\vee b ) ) )\vee c ) ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\wedge a\ne a\qquad\mbox{cnf}(\mbox{prove\_normal\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LAT018-1.p} E3 holds in Ortholattices.

Prove that from ortholattice axioms, one can derive equation E3.




$\mbox{include}(\mbox{'Axioms/LAT003-0.ax'})$

$( ( ( a'\wedge b )\vee ( a'\wedge b' ) )\vee ( a\wedge ( a'\vee b ) ) )'\vee ( a'\vee b )\ne n_1\qquad\mbox{cnf}(\mbox{prove\_e}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LAT175-1.p} Huntington equation H79\_dual implies H32






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$x\vee ( y\wedge ( z\vee ( x\wedge u ) ) ) = x\vee ( ( x\vee ( y\wedge ( x\vee z ) ) )\wedge ( z\vee u ) )\qquad\mbox{cnf}(\mbox{equation\_H79\_dual},\mbox{axiom})$

$a\wedge ( b\vee ( a\wedge ( c\wedge d ) ) )\ne a\wedge ( b\vee ( c\wedge ( ( a\wedge d )\vee ( b\wedge d ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_H}_{32},\mbox{negated\_conjecture})$

\medskip
\end{document}