% problem-set/geometry/tarski/t2.ver1.clauses % created : 07/16/86 % revised : 07/20/89 % description : % Theorem T2: x is between x and y. % representation : % % declare_predicates(4,[L]). % declare_predicates(3,[B,C]). % declare_predicates(2,[EQUAL]). % declare_functions(6,[cont]). % declare_functions(5,[op,euc1,euc2]). % declare_functions(4,[ext]). % declare_constants([C1,C2,C3,a,b,c,d,e]). % declare_variables([u,v,w,x,y,z,v1,v2,v3,v4,v5,x1,y1,z1]). % % Note that skolem functions have been renamed to agree with the second % axiom set's names, for purposed of consistency and readability. Also, % the weakened form of the continuity axiom, axiom A13', is used. % % B(x,y,z) : y is between x and z. % L(x1,y1,x2,y2) : the distance from x1 to y1 equals that from x2 to y2. % C1, C2, C3 : Skolem constants arising from the Lower Dimension Axiom (A11) % op : Skolem function arising from Outer Pasch Axiom (A7) % euc1 : Skolem function arising from Euclid's Axiom (A8) % euc2 : Skolem function arising from Euclid's Axiom (A8) % ext : Skolem function arising from the Segment Construction Axiom (A10) % cont : Skolem function arising from the Weakened form of the Elementary % Continuity Axiom (A13') % A2 : Transitivity axiom for betweenness -B(x,y,v) | -B(y,z,v) | B(x,y,z). % A3 : Connectivity axiom for betweenness -B(x,y,z) | -B(x,y,v) | EQUAL(x,y) | B(x,z,v) | B(x,v,z). % A6 : Transitivity axiom for equidistance -L(x,y,z,v) | -L(x,y,v2,w) | L(z,v,v2,w). % A7 : Outer Pasch Axiom -B(x,w,v) | -B(y,v,z) | B(x,op(w,x,y,z,v),y). -B(x,w,v) | -B(y,v,z) | B(z,w,op(w,x,y,z,v)). % A8 : Euclid's Axiom % B(u,v,euc1(u,v,w,x,y)) | -B(u,w,y) | -B(v,w,x) | EQ(u,w). % B(u,x,euc2(u,v,w,x,y)) | -B(u,w,y) | -B(v,w,x) | EQ(u,w). % B(euc1(u,v,w,x,y),y,euc2(u,v,w,x,y)) | -B(u,w,y) | -B(v,w,x) | EQ(x,v). % A9 : Five Segment Axiom -L(x,y,x1,y1) | -L(y,z,y1,z1) | -L(x,v,x1,v1) | -L(y,v,y1,v1) | -B(x,y,z) | -B(x1,y1,z1) | EQUAL(x,y) | L(z,v,z1,v1). % A11 : Lower Dimension Axiom -B(C1,C2,C3). -B(C2,C3,C1). -B(C3,C1,C2). % A12 : Upper Dimension Axiom % -L(x,w,x,v) | -L(y,w,y,v) | -L(z,w,z,v) | EQUAL(w,v) | B(x,y,z) | % B(y,z,x) | B(z,x,y). % Definition of Collinearity % -C(x,y,z) | B(x,y,z) | B(y,x,z) | B(x,z,y). % -B(x,y,z) | C(x,y,z). % -B(y,x,z) | C(x,y,z). % -B(x,z,y) | C(x,y,z). % Equality axioms EQUAL(x,x). -EQUAL(x,y) | EQUAL(y,x). -EQUAL(x,y) | -EQUAL(y,z) | EQUAL(x,z). % Equality Substitution Axioms -EQUAL(x,y) | -B(x,w,z) | B(y,w,z). -EQUAL(x,y) | -B(w,x,z) | B(w,y,z). -EQUAL(x,y) | -B(w,z,x) | B(w,z,y). % -EQUAL(x,y) | -C(x,w,z) | C(y,w,z). % -EQUAL(x,y) | -C(w,x,z) | C(w,y,z). % -EQUAL(x,y) | -C(w,z,x) | C(w,z,y). -EQUAL(x,y) | -L(x,v,w,z) | L(y,v,w,z). -EQUAL(x,y) | -L(v,x,w,z) | L(v,y,w,z). -EQUAL(x,y) | -L(v,w,x,z) | L(v,w,y,z). -EQUAL(x,y) | -L(v,w,z,x) | L(v,w,y,z). -EQUAL(x,y) | EQUAL(op(x,v1,v2,v3,v4),op(y,v1,v2,v3,v4)). -EQUAL(x,y) | EQUAL(op(v1,x,v2,v3,v4),op(v1,y,v2,v3,v4)). -EQUAL(x,y) | EQUAL(op(v1,v2,x,v3,v4),op(v1,v2,y,v3,v4)). -EQUAL(x,y) | EQUAL(op(v1,v2,v3,x,v4),op(v1,v2,v3,y,v4)). -EQUAL(x,y) | EQUAL(op(v1,v2,v3,v4,x),op(v1,v2,v3,v4,y)). % -EQUAL(x,y) | EQUAL(euc1(x,v1,v2,v3,v4),euc1(y,v1,v2,v3,v4)). % -EQUAL(x,y) | EQUAL(euc1(v1,x,v2,v3,v4),euc1(v1,y,v2,v3,v4)). % -EQUAL(x,y) | EQUAL(euc1(v1,v2,x,v3,v4),euc1(v1,v2,y,v3,v4)). % -EQUAL(x,y) | EQUAL(euc1(v1,v2,v3,x,v4),euc1(v1,v2,v3,y,v4)). % -EQUAL(x,y) | EQUAL(euc1(v1,v2,v3,v4,x),euc1(v1,v2,v3,v4,y)). % -EQUAL(x,y) | EQUAL(euc2(x,v1,v2,v3,v4),euc2(y,v1,v2,v3,v4)). % -EQUAL(x,y) | EQUAL(euc2(v1,x,v2,v3,v4),euc2(v1,y,v2,v3,v4)). % -EQUAL(x,y) | EQUAL(euc2(v1,v2,x,v3,v4),euc2(v1,v2,y,v3,v4)). % -EQUAL(x,y) | EQUAL(euc2(v1,v2,v3,x,v4),euc2(v1,v2,v3,y,v4)). % -EQUAL(x,y) | EQUAL(euc2(v1,v2,v3,v4,x),euc2(v1,v2,v3,v4,y)). -EQUAL(x,y) | EQUAL(ext(x,v1,v2,v3),ext(y,v1,v2,v3)). -EQUAL(x,y) | EQUAL(ext(v1,x,v2,v3),ext(v1,y,v2,v3)). -EQUAL(x,y) | EQUAL(ext(v1,v2,x,v3),ext(v1,v2,y,v3)). -EQUAL(x,y) | EQUAL(ext(v1,v2,v3,x),ext(v1,v2,v3,y)). % -EQUAL(x,y) | EQUAL(cont(x,v1,v2,v3,v4,v5),cont(y,v1,v2,v3,v4,v5)). % -EQUAL(x,y) | EQUAL(cont(v1,x,v2,v3,v4,v5),cont(v1,y,v2,v3,v4,v5)). % -EQUAL(x,y) | EQUAL(cont(v1,v2,x,v3,v4,v5),cont(v1,v2,y,v3,v4,v5)). % -EQUAL(x,y) | EQUAL(cont(v1,v2,v3,x,v4,v5),cont(v1,v2,v3,y,v4,v5)). % -EQUAL(x,y) | EQUAL(cont(v1,v2,v3,v4,x,v5),cont(v1,v2,v3,v4,y,v5)). % -EQUAL(x,y) | EQUAL(cont(v1,v2,v3,v4,v5,x),cont(v1,v2,v3,v4,v5,y)). % T1 : -B(x,y,z) | B(z,y,x). % A1 : Identity axiom for betweenness -B(x,y,x) | EQUAL(x,y). % A4 : Reflexivity axiom for equidistance L(x,y,y,x). % A5 : Identity axiom for equidistance -L(x,y,z,z) | EQUAL(x,y). % A10 : Segment Construction Axiom B(x,y,ext(x,y,w,v)). L(y,ext(x,y,w,v),w,v). % denial of the theorem: -B(a,a,b).