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    \textbf{COL001-0.ax} Type-respecting combinators






$\mbox{apply}(k(x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,k(z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

\medskip\textbf{COL107-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_z}),\mbox{v\_xb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL113-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL016-1.p} Weak fixed point for B, M and L

The weak fixed point property holds for the set P consisting
 of the combinators B, M and L, where ((Bx)y)z = x(yz), (Lx)y
 = x(yy), Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-16.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),w)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-1.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL062-3.p} Find combinator equivalent to C from B and T

Construct from B and T alone a combinator that behaves as the
 combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Cx)y)z = (xz)y




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),b)),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{prove\_c\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL116-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_xb},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL122-2.p} Problem about combinators






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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xaa},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_Ortrancl\_\_refl}_0,\mbox{axiom})$

\medskip\textbf{COL051-1.p} Egocentric mocking bird?

There exists a mocking bird. For all birds x and y, there
 exists a bird z that composes x with y for all birds w. Prove
 that there exists a bird x that is fond of itself.




$\mbox{response}(\mbox{mocking\_bird},y) = \mbox{response}(y,y)\qquad\mbox{cnf}(\mbox{mocking\_bird\_exists},\mbox{axiom})$

$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{hypothesis})$

$\mbox{response}(x,x)\ne x\qquad\mbox{cnf}(\mbox{prove\_the\_bird\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{COL061-3.p} Find combinator equivalent to Q1 from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Q1x)y)z = x(zy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),b)),b),x),y),z)\ne \mbox{apply}(x,\mbox{apply}(z,y))\qquad\mbox{cnf}(\mbox{prove\_q1\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL034-1.p} Strong fixed point for B, M, and T

The strong fixed point property holds for the set
 P consisting of the combinators B, M, and T, where ((Bx)y)z
 = x(yz), Mx = xx, (Tx)y = yx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL102-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_za},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_za},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocontract\_OAp1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Or\_\_into\_\_rtrancl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

\medskip\textbf{COL113-2.p} Problem about combinators






$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

\medskip\textbf{COL084-1.p} Compatible Birds, part 2






$\mbox{response}(\mbox{mocking\_bird},a) = \mbox{response}(a,a)\qquad\mbox{cnf}(\mbox{mocking\_bird\_exists},\mbox{axiom})$

$\mbox{response}(a\circ b,c) = \mbox{response}(a,\mbox{response}(b,c))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{hypothesis})$

$\mbox{response}(b,b)\ne a\qquad\mbox{cnf}(\mbox{prove\_birds\_are\_compatible}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL033-1.p} Strong fixed point for B, M and L

The strong fixed point property holds for the set
 P consisting of the combinators B, M and L, where ((Bx)y)z
 = x(yz), (Lx)y = x(yy), Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL011-1.p} Weak fixed point for O and Q1

The weak fixed point property holds for the set P consisting
 of the combinators O and Q1, where (Ox)y = y(xy), ((Q1x)y)z
 = x(zy).




$\mbox{apply}(\mbox{apply}(o,x),y) = \mbox{apply}(y,\mbox{apply}(x,y))\qquad\mbox{cnf}(\mbox{o\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q_1,x),y),z) = \mbox{apply}(x,\mbox{apply}(z,y))\qquad\mbox{cnf}(\mbox{q1\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL057-1.p} Strong fixed point for S, B, C, and I

The strong fixed point property holds for the set
 P consisting of the combinators S, B, C, and I, where
 ((Sx)y)z = (xz)(yz), ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and
 Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL090-3.p} i\_contract\_E






$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(p,q),r)\in \mbox{contract} \ \Rightarrow \ ( \mbox{ap\_contractE\_c}_1(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_2(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_3(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_4(p,q,r) )\qquad\mbox{cnf}(\mbox{ap\_contractE}_1,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ r\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_2,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_3,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{combK},r)\qquad\mbox{cnf}(\mbox{ap\_contractE}_4,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1p}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_5,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1q}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_6,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_7,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{ap\_contractE\_sk1p}(p,q,r),q),\mbox{comb\_app}(\mbox{ap\_contractE\_sk1q}(p,q,r),q))\qquad\mbox{cnf}(\mbox{ap\_contractE}_8,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},\mbox{ap\_contractE\_sk1p}(p,q,r)),\mbox{ap\_contractE\_sk1q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_9,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ \mbox{pair}(p,\mbox{ap\_contractE\_sk2q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{10},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{11},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{ap\_contractE\_sk2q}(p,q,r),q)\qquad\mbox{cnf}(\mbox{ap\_contractE}_{12},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ \mbox{pair}(q,\mbox{ap\_contractE\_sk3q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{13},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ p\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{14},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(p,\mbox{ap\_contractE\_sk3q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_{15},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combK},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{k\_contractE},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combS},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{s\_contractE},\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},\mbox{combK}),\mbox{combK}),r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{i\_contract\_E},\mbox{negated\_conjecture})$

\medskip\textbf{COL006-7.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(s,k))))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-1.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x))\ne \mbox{apply}(\mbox{apply}(h(x),g(x)),f(x))\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-3.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),b)),\mbox{apply}(\mbox{apply}(b,b),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL098-1.p} diamond\_strip\_lemmaD\_2c1

[rule\_format]:[| diamond(r); $<$x, y$>$ : r$\wedge$+ |]




$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{pair}(a,b)\in r \ \Rightarrow \ \mbox{pair}(a,b)\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{r\_into\_trancl},\mbox{axiom})$

$\mbox{trans}(\mbox{trancl}(r))\qquad\mbox{cnf}(\mbox{trans\_trancl},\mbox{axiom})$

$( \mbox{trans}(r)\mbox{ and }\mbox{pair}(a,b)\in r\mbox{ and }\mbox{pair}(b,c)\in r ) \ \Rightarrow \ \mbox{pair}(a,c)\in r\qquad\mbox{cnf}(\mbox{transD},\mbox{axiom})$

$( \mbox{pair}(x,y)\in r\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(y,\mbox{sk}_1(x,y,\mbox{yP}))\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_1,\mbox{hypothesis})$

$( \mbox{pair}(x,y)\in r\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(\mbox{yP},\mbox{sk}_1(x,y,\mbox{yP}))\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_2,\mbox{hypothesis})$

$\mbox{pair}(x,y)\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_3,\mbox{hypothesis})$

$\mbox{pair}(y,z)\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_4,\mbox{hypothesis})$

$\mbox{pair}(x,\mbox{yP})\in r \ \Rightarrow \ \mbox{pair}(\mbox{yP},\mbox{sk}_2(\mbox{yP}))\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_5,\mbox{hypothesis})$

$\mbox{pair}(x,\mbox{yP})\in r \ \Rightarrow \ \mbox{pair}(y,\mbox{sk}_2(\mbox{yP}))\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2h}_6,\mbox{hypothesis})$

$\mbox{pair}(x,\mbox{sk}_3)\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2c}_1,\mbox{negated\_conjecture})$

$\mbox{pair}(\mbox{sk}_3,\mbox{zA})\in \mbox{trancl}(r) \ \Rightarrow \ \neg\,\mbox{pair}(z,\mbox{zA})\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD\_2c}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL014-1.p} Weak fixed point for L and O

The weak fixed point property holds for the set P consisting
 of the combinators L and O, where (Lx)y = x(yy), (Ox)y
 = y(xy).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(o,x),y) = \mbox{apply}(y,\mbox{apply}(x,y))\qquad\mbox{cnf}(\mbox{o\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL060-3.p} Find combinator equivalent to Q from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,b)),b)),t),x),y),z)\ne \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{prove\_q\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL046-1.p} Strong fixed point for B, M and S

The strong fixed point property holds for the set
 P consisting of the combinators B, M and S, where ((Sx)y)z
 = (xz)(yz), ((Bx)y)z = x(yz), Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-18.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(b,w))),\mbox{apply}(\mbox{apply}(b,b),b))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL042-4.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w_1)),b))),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL006-1.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-8.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(b,w))),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL101-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL001-2.p} Weak fixed point for S and K

The weak fixed point property holds for the set P consisting
 of the combinators S and K alone, where ((Sx)y)z = (xz)(yz)
 and (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i)) = \mbox{apply}(x,\mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i)))\qquad\mbox{cnf}(\mbox{sb\_property},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL039-1.p} Strong fixed point for B, O, and M

The strong fixed point property holds for the set
 P consisting of the combinators B, O, and M, where ((Bx)y)z
 = x(yz), Mx = xx, (Ox)y = y(xy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(o,x),y) = \mbox{apply}(y,\mbox{apply}(x,y))\qquad\mbox{cnf}(\mbox{o\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL053-1.p} An Exercise in Composition

For all birds x and y, there exists a bird z that composes
 x with y for all birds w. Prove that for all birds x, y, and
 z, there exists a bird u such that for all w, uw = x(y(zw)).




$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{axiom})$

$\mbox{response}(u,f(u))\ne \mbox{response}(a,\mbox{response}(b,\mbox{response}(c,f(u))))\qquad\mbox{cnf}(\mbox{prove\_bird\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{COL006-2.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(s,k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k))))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL004-3.p} Find combinator equivalent to U from S and K.

Construct from S and K alone a combinator that behaves as the
 combinator U does, where ((Sx)y)z = (xz)(yz), (Kx)y = x,
 (Ux)y = y((xx)y).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k))),x),y)\ne \mbox{apply}(y,\mbox{apply}(\mbox{apply}(x,x),y))\qquad\mbox{cnf}(\mbox{prove\_u\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL099-1.p} diamond\_trancl\_1c1

diamond(r) ==$>$ diamond(r$\wedge$+)




$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{pair}(a,b)\in r \ \Rightarrow \ \mbox{pair}(a,b)\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{r\_into\_trancl},\mbox{axiom})$

$\mbox{trans}(\mbox{trancl}(r))\qquad\mbox{cnf}(\mbox{trans\_trancl},\mbox{axiom})$

$( \mbox{trans}(r)\mbox{ and }\mbox{pair}(a,b)\in r\mbox{ and }\mbox{pair}(b,c)\in r ) \ \Rightarrow \ \mbox{pair}(a,c)\in r\qquad\mbox{cnf}(\mbox{transD},\mbox{axiom})$

$( \mbox{diamond}(r)\mbox{ and }\mbox{pair}(x,y)\in \mbox{trancl}(r)\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(\mbox{yP},\mbox{diamond\_strip\_lemmaD\_sk}_1(x,y,\mbox{yP},r))\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD}_1,\mbox{axiom})$

$( \mbox{diamond}(r)\mbox{ and }\mbox{pair}(x,y)\in \mbox{trancl}(r)\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(y,\mbox{diamond\_strip\_lemmaD\_sk}_1(x,y,\mbox{yP},r))\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD}_2,\mbox{axiom})$

$\mbox{diamond}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_1h}_1,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{ya})\in r\qquad\mbox{cnf}(\mbox{diamond\_trancl\_1h}_2,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{yp})\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_1c}_1,\mbox{negated\_conjecture})$

$\mbox{pair}(\mbox{ya},z)\in \mbox{trancl}(r) \ \Rightarrow \ \neg\,\mbox{pair}(\mbox{yp},z)\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_1c}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL120-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_3\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_4\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_5\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_6\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_1 = \mbox{v\_comb1\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_2 = \mbox{v\_comb2\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_1,\mbox{axiom})$

$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_0,\mbox{axiom})$

$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL003-2.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,w),\mbox{apply}(\mbox{apply}(b,b),b))))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL066-2.p} Find combinator equivalent to P from B, Q and W

Construct from B, Q and W alone a combinator that behaves as
 the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z =
 y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z)




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(q,q),\mbox{apply}(w,\mbox{apply}(q,\mbox{apply}(q,q)))),x),y),y),z)\ne \mbox{apply}(\mbox{apply}(x,y),\mbox{apply}(\mbox{apply}(x,y),z))\qquad\mbox{cnf}(\mbox{prove\_p\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL116-2.p} Problem about combinators






$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_xb},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

\medskip\textbf{COL006-5.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(s,k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k)))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-12.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,w),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL029-1.p} Strong fixed point for U

The strong fixed point property holds for the set
 P consisting of the combinator U, where (Ux)y = y((xx)y).




$\mbox{apply}(\mbox{apply}(u,x),y) = \mbox{apply}(y,\mbox{apply}(\mbox{apply}(x,x),y))\qquad\mbox{cnf}(\mbox{u\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL036-1.p} Strong fixed point for B, S, and T

The strong fixed point property holds for the set
 P consisting of the combinators B, S, and T, where ((Sx)y)z
 = (xz)(yz), ((Bx)y)z = x(yz), (Tx)y = yx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL112-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL064-4.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,b),b)),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL069-1.p} Strong fixed point for B and L

The strong fixed point property holds for the set
 P consisting of the combinators B and L, where ((Bx)y)z
 = x(yz), (Lx)y = x(yy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL066-1.p} Find combinator equivalent to P from B, Q and W

Construct from B, Q and W alone a combinator that behaves as
 the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z =
 y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z)




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),g(x)),h(x))\ne \mbox{apply}(\mbox{apply}(f(x),g(x)),\mbox{apply}(\mbox{apply}(f(x),g(x)),h(x)))\qquad\mbox{cnf}(\mbox{prove\_p\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL115-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_z}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL002-4.p} Weak fixed point for S, B, C, and I

The weak fixed point property holds for the set P consisting
 of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{weak\_fixed\_point}(x) = \mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i))\qquad\mbox{cnf}(\mbox{weak\_fixed\_point},\mbox{axiom})$

$\mbox{weak\_fixed\_point}(\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{weak\_fixed\_point}(\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_weak\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL078-1.p} Abst Abst Abst Abst = k(k(id))






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},\mbox{abstraction}),\mbox{abstraction}),\mbox{abstraction})\ne k(k(\mbox{identity}))\qquad\mbox{cnf}(\mbox{prove\_TRC1b},\mbox{negated\_conjecture})$

\medskip\textbf{COL015-1.p} Weak fixed point for Q and M

The weak fixed point property holds for the set P consisting
 of the combinators Q and M, where Mx = xx, ((Qx)y)z = y(xz).




$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL043-2.p} Strong fixed point for B and H

The strong fixed point property holds for the set
 P consisting of the combinators B and H, where ((Bx)y)z
 = x(yz), ((Hx)y)z = ((xy)z)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(h,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),z),y)\qquad\mbox{cnf}(\mbox{h\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(h,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,h),\mbox{apply}(b,b))),\mbox{apply}(h,\mbox{apply}(\mbox{apply}(b,h),\mbox{apply}(b,b))))),h)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL067-1.p} Strong fixed point for B and S

The strong fixed point property holds for the set
 P consisting of the combinators B and S, where ((Sx)y)z
 = (xz)(yz), ((Bx)y)z = x(yz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL006-3.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k)),\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k))))))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL114-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x\_H}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_2}_0,\mbox{axiom})$

\medskip\textbf{COL009-1.p} Weak fixed point for B and L2

The weak fixed point property holds for the set P consisting
 of the combinators B and L2, where ((Bx)y)z = x(yz), (L2x)y
 = y(xx).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l_2,x),y) = \mbox{apply}(y,\mbox{apply}(x,x))\qquad\mbox{cnf}(\mbox{l2\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL012-1.p} Weak fixed point for U

The weak fixed point property holds for the set P consisting
 of the combinator U, where (Ux)y = y((xx)y).




$\mbox{apply}(\mbox{apply}(u,x),y) = \mbox{apply}(y,\mbox{apply}(\mbox{apply}(x,x),y))\qquad\mbox{cnf}(\mbox{u\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL081-2.p} Abst k(k(X)) = k(k(X))






$\mbox{apply}(k(x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,k(z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$k(k(b))\ne \mbox{apply}(\mbox{abstraction},k(k(b)))\qquad\mbox{cnf}(\mbox{prove\_TRC2c},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-5.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,b),b))),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL118-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_ya},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{COL108-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_ya},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{v\_xba},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xba},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{COL003-7.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(b,w))),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL112-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_2}_0,\mbox{axiom})$

\medskip\textbf{COL062-2.p} Find combinator equivalent to C from B and T

Construct from B and T alone a combinator that behaves as the
 combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Cx)y)z = (xz)y




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,b),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{prove\_c\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL121-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_3\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_4\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_5\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_6\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_1 = \mbox{v\_comb1\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_2 = \mbox{v\_comb2\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_1,\mbox{axiom})$

$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_0,\mbox{axiom})$

$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_ya},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_xaa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xaa},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{COL055-1.p} Happy Birds

There exists a bird which is fond of some other bird. Prove
 that any bird that is fond of at least one bird must
 be happy.




$\mbox{response}(a,b) = b\qquad\mbox{cnf}(\mbox{fond\_bird\_exists},\mbox{hypothesis})$

$\mbox{response}(a,z) = w \ \Rightarrow \ \mbox{response}(a,w)\ne z\qquad\mbox{cnf}(\mbox{prove\_happiness},\mbox{negated\_conjecture})$

\medskip\textbf{COL118-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_ya},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_w}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_4}_0,\mbox{axiom})$

\medskip\textbf{COL079-1.p} Abst(Abst(Abst X)) = Abst X






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

$\mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},b)))\ne \mbox{apply}(\mbox{abstraction},b)\qquad\mbox{cnf}(\mbox{prove\_TRC2a},\mbox{negated\_conjecture})$

\medskip\textbf{COL090-1.p} i\_contract\_E






$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{comb\_app}(p,q)\in \mbox{comb} \ \Rightarrow \ p\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_E}_1,\mbox{axiom})$

$\mbox{comb\_app}(p,q)\in \mbox{comb} \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_E}_2,\mbox{axiom})$

$\mbox{combK}\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_1,\mbox{axiom})$

$\mbox{combS}\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_2,\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb} ) \ \Rightarrow \ \mbox{comb\_app}(p,q)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_3,\mbox{axiom})$

$a\in \mbox{comb} \ \Rightarrow \ \mbox{pair}(a,a)\in \mbox{rtrancl}(\mbox{contract})\qquad\mbox{cnf}(\mbox{reduction\_refl},\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb} ) \ \Rightarrow \ \mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combK},p),q),p)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{contract\_K},\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb}\mbox{ and }r\in \mbox{comb} ) \ \Rightarrow \ \mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},p),q),r),\mbox{comb\_app}(\mbox{comb\_app}(p,r),\mbox{comb\_app}(q,r)))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{contract\_S},\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(p,q),r)\in \mbox{contract} \ \Rightarrow \ ( \mbox{ap\_contractE\_c}_1(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_2(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_3(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_4(p,q,r) )\qquad\mbox{cnf}(\mbox{ap\_contractE}_1,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ r\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_2,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_3,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{combK},r)\qquad\mbox{cnf}(\mbox{ap\_contractE}_4,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1p}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_5,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1q}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_6,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_7,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{ap\_contractE\_sk1p}(p,q,r),q),\mbox{comb\_app}(\mbox{ap\_contractE\_sk1q}(p,q,r),q))\qquad\mbox{cnf}(\mbox{ap\_contractE}_8,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},\mbox{ap\_contractE\_sk1p}(p,q,r)),\mbox{ap\_contractE\_sk1q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_9,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ \mbox{pair}(p,\mbox{ap\_contractE\_sk2q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{10},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{11},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{ap\_contractE\_sk2q}(p,q,r),q)\qquad\mbox{cnf}(\mbox{ap\_contractE}_{12},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ \mbox{pair}(q,\mbox{ap\_contractE\_sk3q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{13},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ p\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{14},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(p,\mbox{ap\_contractE\_sk3q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_{15},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combK},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{k\_contractE},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combS},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{s\_contractE},\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},\mbox{combK}),\mbox{combK}),r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{i\_contract\_E},\mbox{negated\_conjecture})$

\medskip\textbf{COL121-2.p} Problem about combinators






$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_0,\mbox{axiom})$

$( \mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{c\_Comb\_Odiamond\_\_strip\_\_lemmaE\_}_1(\mbox{v\_r},\mbox{v\_y},\mbox{v\_y\_H},\mbox{t\_a}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Odiamond\_\_strip\_\_lemmaE}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Or\_\_into\_\_rtrancl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Odiamond}(\mbox{v\_r},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_xaa},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xaa},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{COL076-2.p} Lemma 2 for showing the unsatisfiable variant of TRC

Searching for the self-referential combinator with the
 property s = eq $<$k s,k p2$>$.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(f,x),y) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{diagonal\_combinator},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{eq},\mbox{pair}(\mbox{apply}(k,y),\mbox{apply}(k,\mbox{projection}_2)))\qquad\mbox{cnf}(\mbox{prove\_self\_referential},\mbox{negated\_conjecture})$

\medskip\textbf{COL043-3.p} Strong fixed point for B and H

The strong fixed point property holds for the set
 P consisting of the combinators B and H, where ((Bx)y)z
 = x(yz), ((Hx)y)z = ((xy)z)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(h,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),z),y)\qquad\mbox{cnf}(\mbox{h\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(h,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,h),\mbox{apply}(b,b))),\mbox{apply}(h,\mbox{apply}(\mbox{apply}(b,h),\mbox{apply}(b,b))))),h)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL100-1.p} diamond\_trancl\_2c1

diamond(r) ==$>$ diamond(r$\wedge$+)




$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{pair}(a,b)\in r \ \Rightarrow \ \mbox{pair}(a,b)\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{r\_into\_trancl},\mbox{axiom})$

$\mbox{trans}(\mbox{trancl}(r))\qquad\mbox{cnf}(\mbox{trans\_trancl},\mbox{axiom})$

$( \mbox{trans}(r)\mbox{ and }\mbox{pair}(a,b)\in r\mbox{ and }\mbox{pair}(b,c)\in r ) \ \Rightarrow \ \mbox{pair}(a,c)\in r\qquad\mbox{cnf}(\mbox{transD},\mbox{axiom})$

$( \mbox{diamond}(r)\mbox{ and }\mbox{pair}(x,y)\in \mbox{trancl}(r)\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(\mbox{yP},\mbox{diamond\_strip\_lemmaD\_sk}_1(x,y,\mbox{yP},r))\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD}_1,\mbox{axiom})$

$( \mbox{diamond}(r)\mbox{ and }\mbox{pair}(x,y)\in \mbox{trancl}(r)\mbox{ and }\mbox{pair}(x,\mbox{yP})\in r ) \ \Rightarrow \ \mbox{pair}(y,\mbox{diamond\_strip\_lemmaD\_sk}_1(x,y,\mbox{yP},r))\in r\qquad\mbox{cnf}(\mbox{diamond\_strip\_lemmaD}_2,\mbox{axiom})$

$\mbox{diamond}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2h}_1,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{ya})\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2h}_2,\mbox{hypothesis})$

$\mbox{pair}(\mbox{ya},z)\in r\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2h}_3,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{yP})\in \mbox{trancl}(r) \ \Rightarrow \ \mbox{pair}(\mbox{ya},\mbox{sk}_1(\mbox{yP}))\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2h}_4,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{yP})\in \mbox{trancl}(r) \ \Rightarrow \ \mbox{pair}(\mbox{yP},\mbox{sk}_1(\mbox{yP}))\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2h}_5,\mbox{hypothesis})$

$\mbox{pair}(y,\mbox{yp})\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2c}_1,\mbox{negated\_conjecture})$

$\mbox{pair}(z,\mbox{zA})\in \mbox{trancl}(r) \ \Rightarrow \ \neg\,\mbox{pair}(\mbox{yp},\mbox{zA})\in \mbox{trancl}(r)\qquad\mbox{cnf}(\mbox{diamond\_trancl\_2c}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL117-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_y\_Ha},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x\_H}),\mbox{v\_y\_Ha}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{COL104-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_za},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_y}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_za}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_y}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocontract\_OAp2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Or\_\_into\_\_rtrancl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

\medskip\textbf{COL111-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

\medskip\textbf{COL119-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x\_H}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_y\_H}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x\_H}),\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_y\_H}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{COL091-1.p} k1\_contractD\_c1

comb.app(K, p) -1$\to$ r ==$>$ EX q. r = comb.app(K, q) \& p -1$\to$ q




$\mbox{combK}\ne \mbox{combS}\qquad\mbox{cnf}(\mbox{k\_s},\mbox{axiom})$

$\mbox{combK}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{k\_app},\mbox{axiom})$

$\mbox{combS}\ne \mbox{comb\_app}(p,q)\qquad\mbox{cnf}(\mbox{s\_app},\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ p_1 = p_2\qquad\mbox{cnf}(\mbox{app\_app}_1,\mbox{axiom})$

$\mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2) \ \Rightarrow \ q_1 = q_2\qquad\mbox{cnf}(\mbox{app\_app}_2,\mbox{axiom})$

$( p_1 = p_2\mbox{ and }q_1 = q_2 ) \ \Rightarrow \ \mbox{comb\_app}(p_1,q_1) = \mbox{comb\_app}(p_2,q_2)\qquad\mbox{cnf}(\mbox{app\_app}_3,\mbox{axiom})$

$\mbox{comb\_app}(p,q)\in \mbox{comb} \ \Rightarrow \ p\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_E}_1,\mbox{axiom})$

$\mbox{comb\_app}(p,q)\in \mbox{comb} \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_E}_2,\mbox{axiom})$

$\mbox{combK}\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_1,\mbox{axiom})$

$\mbox{combS}\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_2,\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb} ) \ \Rightarrow \ \mbox{comb\_app}(p,q)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{comb\_intros}_3,\mbox{axiom})$

$a\in \mbox{comb} \ \Rightarrow \ \mbox{pair}(a,a)\in \mbox{rtrancl}(\mbox{contract})\qquad\mbox{cnf}(\mbox{reduction\_refl},\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb} ) \ \Rightarrow \ \mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combK},p),q),p)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{contract\_K},\mbox{axiom})$

$( p\in \mbox{comb}\mbox{ and }q\in \mbox{comb}\mbox{ and }r\in \mbox{comb} ) \ \Rightarrow \ \mbox{pair}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},p),q),r),\mbox{comb\_app}(\mbox{comb\_app}(p,r),\mbox{comb\_app}(q,r)))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{contract\_S},\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(p,q),r)\in \mbox{contract} \ \Rightarrow \ ( \mbox{ap\_contractE\_c}_1(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_2(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_3(p,q,r)\mbox{ or }\mbox{ap\_contractE\_c}_4(p,q,r) )\qquad\mbox{cnf}(\mbox{ap\_contractE}_1,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ r\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_2,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_3,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_1(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{combK},r)\qquad\mbox{cnf}(\mbox{ap\_contractE}_4,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1p}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_5,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ \mbox{ap\_contractE\_sk1q}(p,q,r)\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_6,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_7,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{ap\_contractE\_sk1p}(p,q,r),q),\mbox{comb\_app}(\mbox{ap\_contractE\_sk1q}(p,q,r),q))\qquad\mbox{cnf}(\mbox{ap\_contractE}_8,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_2(p,q,r) \ \Rightarrow \ p = \mbox{comb\_app}(\mbox{comb\_app}(\mbox{combS},\mbox{ap\_contractE\_sk1p}(p,q,r)),\mbox{ap\_contractE\_sk1q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_9,\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ \mbox{pair}(p,\mbox{ap\_contractE\_sk2q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{10},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ q\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{11},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_3(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(\mbox{ap\_contractE\_sk2q}(p,q,r),q)\qquad\mbox{cnf}(\mbox{ap\_contractE}_{12},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ \mbox{pair}(q,\mbox{ap\_contractE\_sk3q}(p,q,r))\in \mbox{contract}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{13},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ p\in \mbox{comb}\qquad\mbox{cnf}(\mbox{ap\_contractE}_{14},\mbox{axiom})$

$\mbox{ap\_contractE\_c}_4(p,q,r) \ \Rightarrow \ r = \mbox{comb\_app}(p,\mbox{ap\_contractE\_sk3q}(p,q,r))\qquad\mbox{cnf}(\mbox{ap\_contractE}_{15},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combK},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{k\_contractE},\mbox{axiom})$

$\neg\,\mbox{pair}(\mbox{combS},r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{s\_contractE},\mbox{axiom})$

$\mbox{pair}(\mbox{comb\_app}(\mbox{combK},p),r)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{k1\_contractD\_h}_1,\mbox{hypothesis})$

$r = \mbox{comb\_app}(\mbox{combK},q) \ \Rightarrow \ \neg\,\mbox{pair}(p,q)\in \mbox{contract}\qquad\mbox{cnf}(\mbox{k1\_contractD\_c}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL061-2.p} Find combinator equivalent to Q1 from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Q1x)y)z = x(zy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),\mbox{apply}(\mbox{apply}(b,b),b)),x),y),z)\ne \mbox{apply}(x,\mbox{apply}(z,y))\qquad\mbox{cnf}(\mbox{prove\_q1\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-10.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w)),\mbox{apply}(\mbox{apply}(b,b),b))))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL122-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_3\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_4\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_5\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_6\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_1 = \mbox{v\_comb1\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_2 = \mbox{v\_comb2\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xaa},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_W},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{COL001-1.p} Weak fixed point for S and K

The weak fixed point property holds for the set P consisting
 of the combinators S and K alone, where ((Sx)y)z = (xz)(yz)
 and (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-13.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,w),b)),b))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL078-2.p} Abst Abst Abst Abst = k(k(id))






$\mbox{apply}(k(x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,k(z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},\mbox{abstraction}),\mbox{abstraction}),\mbox{abstraction})\ne k(k(\mbox{identity}))\qquad\mbox{cnf}(\mbox{prove\_TRC1b},\mbox{negated\_conjecture})$

\medskip\textbf{COL114-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x\_H}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL058-3.p} If there's a lark, then there's an egocentric bird.

Suppose we are given a forest that conrtains a lark, and
 we are not given any other information. Prove that at least
 one bird in the forest must be egocentric.




$\mbox{response}(\mbox{response}(\mbox{lark},x_1),x_2) = \mbox{response}(x_1,\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_exists},\mbox{axiom})$

$\mbox{response}(\mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{lark}),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))),\mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{lark}),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))))\ne \mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{lark}),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})))\qquad\mbox{cnf}(\mbox{prove\_the\_bird\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{COL075-1.p} Lemma 1 for showing the unsatisfiable variant of TRC

Searching for a diagonal combinator F with the property
 f X Y = X X.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(y,b(y)),c(y))\ne \mbox{apply}(b(y),b(y))\qquad\mbox{cnf}(\mbox{prove\_diagonal\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL105-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_xb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

\medskip\textbf{COL061-1.p} Find combinator equivalent to Q1 from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Q1x)y)z = x(zy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x))\ne \mbox{apply}(f(x),\mbox{apply}(h(x),g(x)))\qquad\mbox{cnf}(\mbox{prove\_q1\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL085-1.p} Happy Birds, part 1






$\mbox{response}(a,b) = b\qquad\mbox{cnf}(\mbox{fond\_bird\_exists},\mbox{hypothesis})$

$\mbox{response}(a,a)\ne b\qquad\mbox{cnf}(\mbox{prove\_happiness}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL017-1.p} Weak fixed point for B, M, and T

The weak fixed point property holds for the set P consisting
 of the combinators B, M, and T, where ((Bx)y)z = x(yz),
 Mx = xx, (Tx)y = yx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL045-1.p} Weak fixed point for B, M and S

The weak fixed point property holds for the set P consisting
 of the combinators B, M and S, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL123-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OK}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_3\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OK}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_4\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_OS}\ne \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_5\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H})\ne \mbox{c\_Comb\_Ocomb\_OS}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Odistinct\_\_6\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_1 = \mbox{v\_comb1\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb}_1,\mbox{v\_comb}_2) = \mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_comb1\_H},\mbox{v\_comb2\_H}) \ \Rightarrow \ \mbox{v\_comb}_2 = \mbox{v\_comb2\_H}\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocomb\_Oinject\_\_iff1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

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

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_W},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{COL062-1.p} Find combinator equivalent to C from B and T

Construct from B and T alone a combinator that behaves as the
 combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Cx)y)z = (xz)y




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x))\ne \mbox{apply}(\mbox{apply}(f(x),h(x)),g(x))\qquad\mbox{cnf}(\mbox{prove\_c\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL058-1.p} If there's a lark, then there's an egocentric bird.

Suppose we are given a forest that conrtains a lark, and
 we are not given any other information. Prove that at least
 one bird in the forest must be egocentric.




$\mbox{response}(\mbox{response}(\mbox{lark},x_1),x_2) = \mbox{response}(x_1,\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_exists},\mbox{axiom})$

$\mbox{response}(x,x)\ne x\qquad\mbox{cnf}(\mbox{prove\_the\_bird\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{COL026-1.p} Weak fixed point for B and W1

The weak fixed point property holds for the set P consisting
 of the combinators B and W1, where ((Bx)y)z = x(yz), (W1x)y
 = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL120-2.p} Problem about combinators






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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_Ortrancl\_\_refl}_0,\mbox{axiom})$

\medskip\textbf{COL052-1.p} A Question on Agreeable Birds

For all birds x and y, there exists a bird z that composes
 x with y for all birds w. Prove that if C is agreeable then
 A is agreeable.




$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{axiom})$

$\mbox{response}(c,\mbox{common\_bird}(x)) = \mbox{response}(x,\mbox{common\_bird}(x))\qquad\mbox{cnf}(\mbox{agreeable}_1,\mbox{hypothesis})$

$\mbox{response}(a,v)\ne \mbox{response}(\mbox{odd\_bird},v)\qquad\mbox{cnf}(\mbox{prove\_a\_is\_agreeable},\mbox{negated\_conjecture})$

$c = a\circ b\qquad\mbox{cnf}(\mbox{c\_composes\_a\_with\_b},\mbox{hypothesis})$

\medskip\textbf{COL074-3.p} Unsatisfiable variant of TRC

If the function symbol K is replaced by the K combinator then
 the resultant system is inconsistent.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

$\mbox{apply}(k,\mbox{projection}_1) = \mbox{k\_projection}_1\qquad\mbox{cnf}(\mbox{k\_projection}_1,\mbox{axiom})$

$\mbox{apply}(k,\mbox{projection}_2) = \mbox{k\_projection}_2\qquad\mbox{cnf}(\mbox{k\_projection}_2,\mbox{axiom})$

$s = \mbox{apply}(\mbox{eq},\mbox{pair}(\mbox{apply}(k,s),\mbox{apply}(k,\mbox{projection}_2)))\qquad\mbox{cnf}(\mbox{self\_referential},\mbox{axiom})$

\medskip\textbf{COL004-1.p} Find combinator equivalent to U from S and K

Construct from S and K alone a combinator that behaves as the
 combinator U does, where ((Sx)y)z = (xz)(yz), (Kx)y
 = x, (Ux)y = y((xx)y).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(z,f(z)),g(z))\ne \mbox{apply}(g(z),\mbox{apply}(\mbox{apply}(f(z),f(z)),g(z)))\qquad\mbox{cnf}(\mbox{prove\_u\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL119-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x\_H}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_2}_0,\mbox{axiom})$

\medskip\textbf{COL003-9.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w)),b))),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL035-1.p} Strong fixed point for W, Q, and L

The strong fixed point property holds for the set
 P consisting of the combinators W, Q, and L, where (Lx)y
 = x(yy), (Wx)y = (xy)y, ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL124-2.p} Problem about combinators






$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_OI},\mbox{v\_z},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OI\_\_contract\_\_E}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Ocontract\_OK}_0,\mbox{axiom})$

\medskip\textbf{COL042-8.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w_1)),b))),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL109-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_yb},\mbox{v\_y\_Ha},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_xb},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x\_H}),\mbox{v\_y\_Ha}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_xb}),\mbox{v\_yb}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x\_H}),\mbox{v\_y\_Ha}),\mbox{v\_xc}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_xb}),\mbox{v\_yb}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xc}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{COL003-19.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w)),b))),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL037-1.p} Strong fixed point for B, S, and C

The strong fixed point property holds for the set
 P consisting of the combinators B, S, and C, where ((Sx)y)z
 = (xz)(yz), ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL002-5.p} Weak fixed point for S, B, C, and I

The weak fixed point property holds for the set P consisting
 of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{weak\_fixed\_point}(x) = \mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x))),\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x)))),\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x))))\qquad\mbox{cnf}(\mbox{weak\_fixed\_point},\mbox{axiom})$

$\mbox{weak\_fixed\_point}(\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{weak\_fixed\_point}(\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_weak\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL027-1.p} Weak fixed point for B and H

The weak fixed point property holds for the set P consisting
 of the combinators B and H, where ((Bx)y)z = x(yz), ((Hx)y)z
 = ((xy)z)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(h,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),z),y)\qquad\mbox{cnf}(\mbox{h\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL042-9.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w_1)),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL030-1.p} Strong fixed point for S and L

The strong fixed point property holds for the set
 P consisting of the combinators S and L, where ((Sx)y)z
 = (xz)(yz), (Lx)y = x(yy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL023-1.p} Weak fixed point for B and N

The weak fixed point property holds for the set P consisting
 of the combinators B and N, where ((Bx)y)z = x(yz), ((Nx)y)z
 = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL086-1.p} Happy Birds, part 2






$\mbox{response}(a,b) = b\qquad\mbox{cnf}(\mbox{fond\_bird\_exists},\mbox{hypothesis})$

$\mbox{response}(a,b)\ne a\qquad\mbox{cnf}(\mbox{prove\_happiness}_2,\mbox{negated\_conjecture})$

\medskip\textbf{COL123-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Or\_\_into\_\_rtrancl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

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

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_W},\mbox{v\_xa}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{t\_a},\mbox{t\_a}),\mbox{v\_r},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{COL005-1.p} Find a model for S and W but not a weak fixed point

The model one is seeking must satisfy S and W and fail
 to satisfy the weak fixed point property, where ((Sx)y)z
 = (xz)(yz), (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_model},\mbox{negated\_conjecture})$

\medskip\textbf{COL110-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_yb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_wa},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_wa}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_8,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_w}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_4}_0,\mbox{axiom})$

\medskip\textbf{COL083-1.p} Compatible Birds, part 1






$\mbox{response}(\mbox{mocking\_bird},a) = \mbox{response}(a,a)\qquad\mbox{cnf}(\mbox{mocking\_bird\_exists},\mbox{axiom})$

$\mbox{response}(a\circ b,c) = \mbox{response}(a,\mbox{response}(b,c))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{hypothesis})$

$\mbox{response}(a,a)\ne b\qquad\mbox{cnf}(\mbox{prove\_birds\_are\_compatible}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL042-2.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(b,w_1))),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-9.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,b)),\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(b,b)))))),n)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL074-1.p} Unsatisfiable variant of TRC

If the function symbol K is replaced by the K combinator then
 the resultant system is inconsistent.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

\medskip\textbf{COL103-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OAp\_\_reduce1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{COL041-1.p} Strong fixed point for B, M, and C

The strong fixed point property holds for the set
 P consisting of the combinators B, M, and C, where ((Bx)y)z
 = x(yz), Mx = xx, ((Cx)y)z = (xz)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL073-1.p} Strong fixed point for B and N1

The strong fixed point property holds for the set
 P consisting of the combinators B and N1, where N1xyz = xyyz,
 ((Bx)y)z = x(yz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(n_1,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),y),z)\qquad\mbox{cnf}(\mbox{n1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-6.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),b)),b)),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,y),x)\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL101-2.p} Problem about combinators






$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_Ortrancl\_\_refl}_0,\mbox{axiom})$

\medskip\textbf{COL048-1.p} Weak fixed point for B, W, and M

The weak fixed point property holds for the set P consisting
 of the combinators B, W, and M, where ((Bx)y)z = x(yz), (Wx)y
 = (xy)y, Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL007-1.p} Weak fixed point for L

The weak fixed point property holds for the set P consisting
 of the combinator L, where (Lx)y = x(yy).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL103-2.p} Problem about combinators






$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_Ortrancl\_\_refl}_0,\mbox{axiom})$

\medskip\textbf{COL080-2.p} Abst(Abst k(X)) = k(X)






$\mbox{apply}(k(x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,k(z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$k(b)\ne \mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},k(b)))\qquad\mbox{cnf}(\mbox{prove\_TRC2b},\mbox{negated\_conjecture})$

\medskip\textbf{COL104-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OAp\_\_reduce1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_za},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_y}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_za}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{COL064-1.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x))\ne \mbox{apply}(\mbox{apply}(h(x),f(x)),g(x))\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL018-1.p} Weak fixed point for W, Q, and L

The weak fixed point property holds for the set P consisting
 of the combinators W, Q, and L, where (Lx)y = x(yy), (Wx)y
 = (xy)y, ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-6.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),w)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-2.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(b,b),t))),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-1.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL022-1.p} Weak fixed point for B, O, and M

The weak fixed point property holds for the set P consisting
 of the combinators B, O, and M, where ((Bx)y)z = x(yz),
 Mx = xx, (Ox)y = y(xy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(o,x),y) = \mbox{apply}(y,\mbox{apply}(x,y))\qquad\mbox{cnf}(\mbox{o\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL087-1.p} Strong fixed point for B and M

The strong fixed point property holds for the set with the
 combinators B and M as a basis, where Bxyz = x(yz) and
 Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{definition\_B},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{definition\_M},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{strong\_fixpoint},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-4.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,b),b)),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,y),x)\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL031-1.p} Strong fixed point for L and O

The strong fixed point property holds for the set
 P consisting of the combinators L and O, where (Lx)y = x(yy),
 (Ox)y = y(xy).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(o,x),y) = \mbox{apply}(y,\mbox{apply}(x,y))\qquad\mbox{cnf}(\mbox{o\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL075-2.p} Lemma 1 for showing the unsatisfiable variant of TRC

Searching for a diagonal combinator F with the property
 f X Y = X X.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(y,b(y)),c(y))\ne \mbox{apply}(b(y),b(y))\qquad\mbox{cnf}(\mbox{prove\_diagonal\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-5.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),\mbox{apply}(\mbox{apply}(b,b),b))),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,y),x)\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL042-7.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(b,w_1))),\mbox{apply}(\mbox{apply}(b,b),b))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL077-1.p} Abst Abst Abst Abst Abst Abst = Id






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},\mbox{abstraction}),\mbox{abstraction}),\mbox{abstraction}),\mbox{abstraction}),\mbox{abstraction})\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_TRC1a},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-4.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,w),b))),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL050-1.p} The Significance of the Mockingbird

There exists a mocking bird. For all birds x and y, there
 exists a bird z that composes x with y for all birds w. Prove
 that every bird is fond of at least one other bird.




$\mbox{response}(\mbox{mocking\_bird},y) = \mbox{response}(y,y)\qquad\mbox{cnf}(\mbox{mocking\_bird\_exists},\mbox{axiom})$

$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{hypothesis})$

$\mbox{response}(a,y)\ne y\qquad\mbox{cnf}(\mbox{prove\_all\_fond\_of\_another},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-10.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,b),t))),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL058-2.p} If there's a lark, then there's an egocentric bird.

Suppose we are given a forest that contains a lark, and
 we are not given any other information. Prove that at least
 one bird in the forest must be egocentric.




$\mbox{response}(\mbox{response}(\mbox{lark},x_1),x_2) = \mbox{response}(x_1,\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_exists},\mbox{axiom})$

$\mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))),\mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark}))))\ne \mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})))),\mbox{response}(\mbox{lark},\mbox{response}(\mbox{lark},\mbox{lark})))\qquad\mbox{cnf}(\mbox{prove\_the\_bird\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{COL008-1.p} Weak fixed point for M and B

The weak fixed point property holds for the set P consisting
 of the combinators M and B, where ((Bx)y)z = x(yz), Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL065-1.p} Find combinator equivalent to G from B and T

Construct from B and T alone a combinator that behaves as the
 combinator G does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 (((Gx)y)z)w = (xw)(yz)




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x)),i(x))\ne \mbox{apply}(\mbox{apply}(f(x),i(x)),\mbox{apply}(g(x),h(x)))\qquad\mbox{cnf}(\mbox{prove\_g\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL002-1.p} Weak fixed point for S, B, C, and I

The weak fixed point property holds for the set P consisting
 of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{fixed\_pt},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL013-1.p} Weak fixed point for S and L

The weak fixed point property holds for the set P consisting
 of the combinators S and L, where ((Sx)y)z = (xz)(yz), (Lx)y
 = x(yy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL006-6.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k)))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k)),\mbox{apply}(k,\mbox{apply}(\mbox{apply}(s,\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,k),k)))))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-5.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,b)),\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(b,b)))))),n)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL056-1.p} Normal Birds

For all birds x and y, there exists a bird z that composes
 x with y for all birds w. Prove that if there exists a happy
 bird then there exists a normal bird.




$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{axiom})$

$\mbox{response}(a,b) = c\qquad\mbox{cnf}(\mbox{a\_to\_b\_responds\_c},\mbox{hypothesis})$

$\mbox{response}(a,c) = b\qquad\mbox{cnf}(\mbox{a\_to\_c\_responds\_b},\mbox{hypothesis})$

$\mbox{response}(w,v)\ne v\qquad\mbox{cnf}(\mbox{prove\_there\_exists\_a\_happy\_bird},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-5.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),w)),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL111-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y\_H},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL079-2.p} Abst(Abst(Abst X)) = Abst X






$\mbox{apply}(k(x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,k(z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},b)))\ne \mbox{apply}(\mbox{abstraction},b)\qquad\mbox{cnf}(\mbox{prove\_TRC2a},\mbox{negated\_conjecture})$

\medskip\textbf{COL070-1.p} Weak fixed point for B and N1

The weak fixed point property holds for the set P consisting
 of the combinators B and N1, where N1xyz = xyyz, ((Bx)y)z
 = x(yz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(n_1,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),y),z)\qquad\mbox{cnf}(\mbox{n1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL042-6.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(b,w_1))),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL124-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OAp\_\_reduce1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_x}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_z},\mbox{v\_y}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OAp\_\_reduce2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_OI},\mbox{c\_Comb\_OI})),\mbox{c\_Comb\_OI},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OKIII\_\_contract1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_OI},\mbox{c\_Comb\_OI})),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}))),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OKIII\_\_contract2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{c\_Comb\_OI}))),\mbox{c\_Comb\_OI},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_OKIII\_\_contract3}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_V},\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_W},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_V},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_W},\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V},\mbox{v\_W}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL071-1.p} Strong fixed point for N and Q

The strong fixed point property holds for the set
 P consisting of the combinators N and Q, where ((Nx)y)z
 = ((xz)y)z, ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-15.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),w)),\mbox{apply}(\mbox{apply}(b,b),b))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL052-2.p} A Question on Agreeable Birds

For all birds x and y, there exists a bird z that composes
 x with y for all birds w. Prove that if C is agreeable then
 A is agreeable.




$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{axiom})$

$\mbox{agreeable}(x) \ \Rightarrow \ \mbox{response}(x,\mbox{common\_bird}(y)) = \mbox{response}(y,\mbox{common\_bird}(y))\qquad\mbox{cnf}(\mbox{agreeable}_1,\mbox{axiom})$

$\mbox{response}(x,z) = \mbox{response}(\mbox{compatible}(x),z) \ \Rightarrow \ \mbox{agreeable}(x)\qquad\mbox{cnf}(\mbox{agreeable}_2,\mbox{axiom})$

$\mbox{agreeable}(c)\qquad\mbox{cnf}(\mbox{c\_is\_agreeable},\mbox{hypothesis})$

$\neg\,\mbox{agreeable}(a)\qquad\mbox{cnf}(\mbox{prove\_a\_is\_agreeable},\mbox{negated\_conjecture})$

$c = a\circ b\qquad\mbox{cnf}(\mbox{c\_composes\_a\_with\_b},\mbox{hypothesis})$

\medskip\textbf{COL064-8.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),b)),\mbox{apply}(\mbox{apply}(b,t),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL076-1.p} Lemma 2 for showing the unsatisfiable variant of TRC

Searching for the self-referential combinator with the
 property s = eq $<$k s,k p2$>$.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{axiom})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(f,x),y) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{diagonal\_combinator},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{eq},\mbox{pair}(\mbox{apply}(k,y),\mbox{apply}(k,\mbox{projection}_2)))\qquad\mbox{cnf}(\mbox{prove\_self\_referential},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-7.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(b,b))),n))),n)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL105-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_xb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL060-1.p} Find combinator equivalent to Q from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(x,f(x)),g(x)),h(x))\ne \mbox{apply}(g(x),\mbox{apply}(f(x),h(x)))\qquad\mbox{cnf}(\mbox{prove\_q\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL066-3.p} Find combinator equivalent to P from B, Q and W

Construct from B, Q and W alone a combinator that behaves as
 the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z =
 y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z)




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,\mbox{apply}(q,\mbox{apply}(q,q)))),q),x),y),y),z)\ne \mbox{apply}(\mbox{apply}(x,y),\mbox{apply}(\mbox{apply}(x,y),z))\qquad\mbox{cnf}(\mbox{prove\_p\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL002-2.p} Weak fixed point for S, B, C, and I

The weak fixed point property holds for the set P consisting
 of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{weak\_sage} = \mbox{apply}(\mbox{fixed\_pt},\mbox{weak\_sage}) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{weak\_sage})\qquad\mbox{cnf}(\mbox{weak\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(b,x)),i)))\qquad\mbox{cnf}(\mbox{prove\_weak\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL042-3.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(b,w_1))),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL047-1.p} Find a model for L and Q but not a strong fixed point

The model one is seeking must satisfy L and Q and fail
 to satisfy the strong fixed point property, where (Lx)y
 = x(yy), ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(l,x),y) = \mbox{apply}(x,\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{l\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_model},\mbox{negated\_conjecture})$

\medskip\textbf{COL117-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_y\_Ha},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x\_H}),\mbox{v\_y\_Ha}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z})),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_3}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_w}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_4}_0,\mbox{axiom})$

\medskip\textbf{COL115-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_z}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_x},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z})),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_3}_0,\mbox{axiom})$

\medskip\textbf{COL002-3.p} Weak fixed point for S, B, C, and I

The weak fixed point property holds for the set P consisting
 of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$\mbox{apply}(i,x) = x\qquad\mbox{cnf}(\mbox{i\_definition},\mbox{axiom})$

$\mbox{weak\_sage} = \mbox{apply}(\mbox{fixed\_pt},\mbox{weak\_sage}) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{weak\_sage})\qquad\mbox{cnf}(\mbox{weak\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x))),\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x)))),\mbox{apply}(s,\mbox{apply}(c,\mbox{apply}(b,x)))))\qquad\mbox{cnf}(\mbox{prove\_weak\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-2.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(b,b),t))),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,y),x)\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL110-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_ya},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_yb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_wa},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_wa}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_ya},\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_ya},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xb}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_w},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_U},\mbox{v\_xaa}(\mbox{v\_U}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_8,\mbox{negated\_conjecture})$

\medskip\textbf{COL006-4.p} Strong fixed point for S and K

The strong fixed point property holds for the set
 P consisting of the combinators S and K alone, where
 ((Sx)y)z = (xz)(yz), (Kx)y = x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,\mbox{apply}(\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(\mbox{apply}(s,k),k)),\mbox{apply}(\mbox{apply}(s,s),\mbox{apply}(s,k))))),\mbox{apply}(\mbox{apply}(s,\mbox{apply}(k,s)),k)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL049-1.p} Strong fixed point for B, W, and M

The strong fixed point property holds for the set
 P consisting of the combinators B, W, and M, where ((Bx)y)z
 = x(yz), (Wx)y = (xy)y, Mx = xx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL106-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OK},\mbox{v\_x}),\mbox{v\_ya}),\mbox{v\_xb},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{COL042-5.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w_1,w_1)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w_1)),\mbox{apply}(\mbox{apply}(b,b),b))))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL102-1.p} Problem about combinators






$\mbox{include}(\mbox{'Axioms/COL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

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

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{v\_r},\mbox{t\_a}),\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Transitive\_\_Closure\_Ortrancl\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_y},\mbox{v\_za},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Ocontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_za},\mbox{v\_z}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Transitive\_\_Closure\_Ortrancl}(\mbox{c\_Comb\_Ocontract},\mbox{tc\_Comb\_Ocomb}),\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{COL042-1.p} Strong fixed point for B and W1

The strong fixed point property holds for the set
 P consisting of the combinators B and W1, where ((Bx)y)z
 = x(yz), (W1x)y = (yx)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w_1,x),y) = \mbox{apply}(\mbox{apply}(y,x),x)\qquad\mbox{cnf}(\mbox{w1\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-2.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),n)),n))),n)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL054-1.p} Compatible Birds

There exists a mockingbird. For all birds x and y, there
 exists a bird z that composes x with y for all birds w. Prove
 that any two birds are compatible.




$\mbox{response}(\mbox{mocking\_bird},y) = \mbox{response}(y,y)\qquad\mbox{cnf}(\mbox{mocking\_bird\_exists},\mbox{axiom})$

$\mbox{response}(x\circ y,w) = \mbox{response}(x,\mbox{response}(y,w))\qquad\mbox{cnf}(\mbox{composer\_exists},\mbox{hypothesis})$

$\mbox{response}(a,x) = y \ \Rightarrow \ \mbox{response}(b,y)\ne x\qquad\mbox{cnf}(\mbox{prove\_birds\_are\_compatible},\mbox{negated\_conjecture})$

\medskip\textbf{COL081-1.p} Abst k(k(X)) = k(k(X))






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$k(k(b))\ne \mbox{apply}(\mbox{abstraction},k(k(b)))\qquad\mbox{cnf}(\mbox{prove\_TRC2c},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-11.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),b)),t)),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL068-1.p} Weak fixed point for B and S

The weak fixed point property holds for the set P consisting
 of the combinators B and S, where ((Sx)y)z (xz)(yz), ((Bx)y)z
 = x(yz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL032-1.p} Strong fixed point for Q and M

The strong fixed point property holds for the set
 P consisting of the combinators Q and M, where Mx = xx,
 ((Qx)y)z = y(xz).




$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(q,x),y),z) = \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{q\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-6.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),n)),n))),n)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-3.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,w),b)),b)))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-3.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(b,b))),n))),n)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL044-4.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}) = \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})) \ \Rightarrow \ \mbox{fixed\_point}(\mbox{strong\_fixed\_point})\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\neg\,\mbox{fixed\_point}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,b)),\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),n))))),n)),b)),b))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-6.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),b)),b)),t),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL074-2.p} Unsatisfiable variant of TRC

If the function symbol K is replaced by the K combinator then
 the resultant system is inconsistent.




$\mbox{apply}(\mbox{apply}(k,x),y) = x\qquad\mbox{cnf}(\mbox{k\_definition},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{projection}_1,\mbox{pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{projection}_1,\mbox{axiom})$

$\mbox{apply}(\mbox{projection}_2,\mbox{pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{projection}_2,\mbox{axiom})$

$\mbox{pair}(\mbox{apply}(\mbox{projection}_1,x),\mbox{apply}(\mbox{projection}_2,x)) = x\qquad\mbox{cnf}(\mbox{pairing},\mbox{axiom})$

$\mbox{apply}(\mbox{pair}(x,y),z) = \mbox{pair}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{pairwise\_application},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{abstraction},x),y),z) = \mbox{apply}(\mbox{apply}(x,\mbox{apply}(k,z)),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{abstraction},\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{eq},\mbox{pair}(x,x)) = \mbox{projection}_1\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$x = y\mbox{ or }\mbox{apply}(\mbox{eq},\mbox{pair}(x,y)) = \mbox{projection}_2\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$\mbox{apply}(x,n(x,y)) = \mbox{apply}(y,n(x,y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$\mbox{projection}_1\ne \mbox{projection}_2\qquad\mbox{cnf}(\mbox{different\_projections},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(f,x),y) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{diagonal\_combinator},\mbox{axiom})$

\medskip\textbf{COL059-1.p} L3 ((lark lark) lark) is not egocentric.






$\mbox{response}(\mbox{response}(\mbox{kestrel},x_1),x_2) = x_1\qquad\mbox{cnf}(\mbox{kestrel\_exists},\mbox{axiom})$

$\mbox{response}(\mbox{response}(\mbox{lark},x_1),x_2) = \mbox{response}(x_1,\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_exists},\mbox{axiom})$

$\mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{lark}),x_1),x_2) = \mbox{response}(\mbox{response}(x_1,x_1),\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_lemma}_1,\mbox{axiom})$

$\mbox{response}(\mbox{response}(\mbox{response}(\mbox{response}(\mbox{lark},\mbox{lark}),\mbox{lark}),x_1),x_2) = \mbox{response}(\mbox{response}(\mbox{response}(x_1,x_1),\mbox{response}(x_1,x_1)),\mbox{response}(x_2,x_2))\qquad\mbox{cnf}(\mbox{lark\_lemma}_2,\mbox{axiom})$

$\mbox{response}(l_2,l_2)\ne l_2\qquad\mbox{cnf}(\mbox{lark\_not\_egocentric},\mbox{axiom})$

$\mbox{response}(\mbox{lark},\mbox{lark}) = l_2\qquad\mbox{cnf}(\mbox{l2\_definition},\mbox{axiom})$

$\mbox{response}(l_2,\mbox{lark}) = l_3\qquad\mbox{cnf}(\mbox{l3\_definition},\mbox{axiom})$

$\mbox{response}(l_3,l_3) = l_3\qquad\mbox{cnf}(\mbox{prove\_l3\_not\_egocentric},\mbox{negated\_conjecture})$

\medskip\textbf{COL080-1.p} Abst(Abst k(X)) = k(X)






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

$\mbox{apply}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{identity\_definition},\mbox{axiom})$

$k(b)\ne \mbox{apply}(\mbox{abstraction},\mbox{apply}(\mbox{abstraction},k(b)))\qquad\mbox{cnf}(\mbox{prove\_TRC2b},\mbox{negated\_conjecture})$

\medskip\textbf{COL024-1.p} Weak fixed point for B, M, and C

The weak fixed point property holds for the set P consisting
 of the combinators B, M, and C, where ((Bx)y)z = x(yz),
 Mx = xx, ((Cx)y)z = (xz)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL063-3.p} Find combinator equivalent to F from B and T

Construct from B and T alone a combinator that behaves as the
 combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Fx)y)z = (zy)x.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,t)),b)),\mbox{apply}(\mbox{apply}(b,b),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,y),x)\qquad\mbox{cnf}(\mbox{prove\_f\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL043-1.p} Strong fixed point for B and H

The strong fixed point property holds for the set
 P consisting of the combinators B and H, where ((Bx)y)z
 = x(yz), ((Hx)y)z = ((xy)z)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(h,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,y),z),y)\qquad\mbox{cnf}(\mbox{h\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL025-1.p} Weak fixed point for B and W

The weak fixed point property holds for the set P consisting
 of the combinators B and W, where ((Bx)y)z = x(yz), (Wx)y
 = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-17.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(b,w))),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL019-1.p} Weak fixed point for B, S, and T

The weak fixed point property holds for the set P consisting
 of the combinators B, S, and T, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), (Tx)y = yx.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL003-20.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,w)),\mbox{apply}(\mbox{apply}(b,b),b)))\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL060-2.p} Find combinator equivalent to Q from B and T

Construct from B and T alone a combinator that behaves as the
 combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Qx)y)z = y(xz).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,b)),\mbox{apply}(\mbox{apply}(b,b),t)),x),y),z)\ne \mbox{apply}(y,\mbox{apply}(x,z))\qquad\mbox{cnf}(\mbox{prove\_q\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL010-1.p} Weak fixed point for B and S2

The weak fixed point property holds for the set P consisting
 of the combinators B and S2, where ((Bx)y)z = x(yz),
 ((S2x)y)z = (xz)(yy).




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(s_2,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,y))\qquad\mbox{cnf}(\mbox{s2\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL082-1.p} Type-respecting combinators






$\mbox{include}(\mbox{'Axioms/COL001-0.ax'})$

\medskip\textbf{COL044-8.p} Strong fixed point for B and N

The strong fixed point property holds for the set
 P consisting of the combinators B and N, where ((Bx)y)z
 = x(yz), ((Nx)y)z = ((xz)y)z.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(n,x),y),z) = \mbox{apply}(\mbox{apply}(\mbox{apply}(x,z),y),z)\qquad\mbox{cnf}(\mbox{n\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(n,\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(b,b)),\mbox{apply}(n,\mbox{apply}(\mbox{apply}(b,b),n))))),n)),b)),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-7.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,b),\mbox{apply}(\mbox{apply}(b,t),t))),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip\textbf{COL038-1.p} Strong fixed point for B, M, and V

The strong fixed point property holds for the set
 P consisting of the combinators B, M, and V, where ((Bx)y)z
 = x(yz), Mx = xx, ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(v,x),y),z) = \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{v\_definition},\mbox{axiom})$

$\mbox{apply}(y,f(y))\ne \mbox{apply}(f(y),\mbox{apply}(y,f(y)))\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL021-1.p} Weak fixed point for B, M, and V

The weak fixed point property holds for the set P consisting
 of the combinators B, M, and V, where ((Bx)y)z = x(yz),
 Mx = xx, ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(m,x) = \mbox{apply}(x,x)\qquad\mbox{cnf}(\mbox{m\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(v,x),y),z) = \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{v\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL109-2.p} Problem about combinators






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_xb},\mbox{v\_x\_H},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_yb},\mbox{v\_y\_Ha},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_xb},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_yb},\mbox{v\_z})),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x\_H}),\mbox{v\_y\_Ha}),\mbox{v\_w}),\mbox{v\_U},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_OS},\mbox{v\_x}),\mbox{v\_y}),\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_z})),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_3}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_z},\mbox{v\_w},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_x},\mbox{v\_z}),\mbox{c\_Comb\_Ocomb\_Oop\_A\_D\_D}(\mbox{v\_y},\mbox{v\_w}),\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}),\mbox{c\_Comb\_Oparcontract},\mbox{tc\_prod}(\mbox{tc\_Comb\_Ocomb},\mbox{tc\_Comb\_Ocomb}))\qquad\mbox{cnf}(\mbox{cls\_Comb\_Oparcontract\_Ointros\_\_4}_0,\mbox{axiom})$

\medskip\textbf{COL003-14.p} Strong fixed point for B and W

The strong fixed point property holds for the set
 P consisting of the combinators B and W alone, where ((Bx)y)z
 = x(yz) and (Wx)y = (xy)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(w,x),y) = \mbox{apply}(\mbox{apply}(x,y),y)\qquad\mbox{cnf}(\mbox{w\_definition},\mbox{axiom})$

$\mbox{strong\_fixed\_point} = \mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,\mbox{apply}(w,w)),\mbox{apply}(\mbox{apply}(b,w),b))),b)\qquad\mbox{cnf}(\mbox{strong\_fixed\_point},\mbox{axiom})$

$\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt})\ne \mbox{apply}(\mbox{fixed\_pt},\mbox{apply}(\mbox{strong\_fixed\_point},\mbox{fixed\_pt}))\qquad\mbox{cnf}(\mbox{prove\_strong\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL020-1.p} Weak fixed point for B, S, and C

The weak fixed point property holds for the set P consisting
 of the combinators B, S, and C, where ((Sx)y)z = (xz)(yz),
 ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(s,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{s\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(c,x),y),z) = \mbox{apply}(\mbox{apply}(x,z),y)\qquad\mbox{cnf}(\mbox{c\_definition},\mbox{axiom})$

$y\ne \mbox{apply}(\mbox{combinator},y)\qquad\mbox{cnf}(\mbox{prove\_fixed\_point},\mbox{negated\_conjecture})$

\medskip\textbf{COL064-9.p} Find combinator equivalent to V from B and T

Construct from B and T alone a combinator that behaves as the
 combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx,
 ((Vx)y)z = (zx)y.




$\mbox{apply}(\mbox{apply}(\mbox{apply}(b,x),y),z) = \mbox{apply}(x,\mbox{apply}(y,z))\qquad\mbox{cnf}(\mbox{b\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(t,x),y) = \mbox{apply}(y,x)\qquad\mbox{cnf}(\mbox{t\_definition},\mbox{axiom})$

$\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(\mbox{apply}(b,\mbox{apply}(t,\mbox{apply}(\mbox{apply}(b,b),t))),\mbox{apply}(\mbox{apply}(b,\mbox{apply}(\mbox{apply}(b,b),t)),t)),x),y),z)\ne \mbox{apply}(\mbox{apply}(z,x),y)\qquad\mbox{cnf}(\mbox{prove\_v\_combinator},\mbox{negated\_conjecture})$

\medskip
\end{document}