% problem-set/geometry/tarski/t13.ver1.in % created : 07/16/86 % revised : 07/29/89 % description : % Theorem T13: If z1, z2, and z3 are each collinear with distinct points % x and y, then z1, z2, and z3 are collinear. % representation : % % declare_predicates(4,[L]). % declare_predicates(3,[B,C]). % declare_predicates(2,[EQ]). % 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') set(hyper_res). set(ur_res). set(order_eq). set(unit_deletion). assign(max_mem,2000). assign(max_seconds,2400). set_skolem([ext(x,y,z,w), op(u,v,w,x,y)]). set(delete_identical_nested_skolem). assign(max_weight,30). list(axioms). % A1 : Identity axiom for betweenness -B(x,y,x) | EQ(x,y). % 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) | B(x,z,v) | B(x,v,z) | EQ(x,y). % A4 : Reflexivity axiom for equidistance L(x,y,y,x). % A5 : Identity axiom for equidistance -L(x,y,z,z) | EQ(x,y). % 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,op(w,x,y,z,v),y) | -B(x,w,v) | -B(y,v,z). B(z,w,op(w,x,y,z,v)) | -B(x,w,v) | -B(y,v,z). % 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(u,w). % A9 : Five Segment Axiom -L(x,y,x1,y1) | -L(y,z,y1,z1) | -L(x,v,x1,v1) | -L(y,v,y1,v1) | L(z,v,z1,v1) | -B(x,y,z) | -B(x1,y1,z1) | EQ(x,y). % A10 : Segment Construction Axiom B(x,y,ext(x,y,w,v)). L(y,ext(x,y,w,v),w,v). % 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) | B(x,y,z) | % B(y,z,x) | B(z,x,y) | EQ(w,v). % A13 : Continuity Axiom (Weak Form) % -L(v,x,v,x1) | -L(v,z,v,z1) | -B(v,x,z) | -B(x,y,z) | % L(v,y,v,cont(x,y,z,x1,z1,v)). % -L(v,x,v,x1) | -L(v,z,v,z1) | -B(v,x,z) | -B(x,y,z) | % B(x1,cont(x,y,z,x1,z1,v),z1). % 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 EQ(x,x). -EQ(x,y) | EQ(y,x). -EQ(x,y) | -EQ(y,z) | EQ(x,z). % Equality Substitution Axioms -B(x,w,z) | B(y,w,z) | -EQ(x,y). -B(w,x,z) | B(w,y,z) | -EQ(x,y). -B(w,z,x) | B(w,z,y) | -EQ(x,y). -C(x,w,z) | C(y,w,z) | -EQ(x,y). -C(w,x,z) | C(w,y,z) | -EQ(x,y). -C(w,z,x) | C(w,z,y) | -EQ(x,y). -L(x,v,w,z) | L(y,v,w,z) | -EQ(x,y). -L(v,x,w,z) | L(v,y,w,z) | -EQ(x,y). -L(v,w,x,z) | L(v,w,y,z) | -EQ(x,y). -L(v,w,z,x) | L(v,w,y,z) | -EQ(x,y). EQ(op(x,v1,v2,v3,v4),op(y,v1,v2,v3,v4)) | -EQ(x,y). EQ(op(v1,x,v2,v3,v4),op(v1,y,v2,v3,v4)) | -EQ(x,y). EQ(op(v1,v2,x,v3,v4),op(v1,v2,y,v3,v4)) | -EQ(x,y). EQ(op(v1,v2,v3,x,v4),op(v1,v2,v3,y,v4)) | -EQ(x,y). EQ(op(v1,v2,v3,v4,x),op(v1,v2,v3,v4,y)) | -EQ(x,y). % EQ(euc1(x,v1,v2,v3,v4),euc1(y,v1,v2,v3,v4)) | -EQ(x,y). % EQ(euc1(v1,x,v2,v3,v4),euc1(v1,y,v2,v3,v4)) | -EQ(x,y). % EQ(euc1(v1,v2,x,v3,v4),euc1(v1,v2,y,v3,v4)) | -EQ(x,y). % EQ(euc1(v1,v2,v3,x,v4),euc1(v1,v2,v3,y,v4)) | -EQ(x,y). % EQ(euc1(v1,v2,v3,v4,x),euc1(v1,v2,v3,v4,y)) | -EQ(x,y). % EQ(euc2(x,v1,v2,v3,v4),euc2(y,v1,v2,v3,v4)) | -EQ(x,y). % EQ(euc2(v1,x,v2,v3,v4),euc2(v1,y,v2,v3,v4)) | -EQ(x,y). % EQ(euc2(v1,v2,x,v3,v4),euc2(v1,v2,y,v3,v4)) | -EQ(x,y). % EQ(euc2(v1,v2,v3,x,v4),euc2(v1,v2,v3,y,v4)) | -EQ(x,y). % EQ(euc2(v1,v2,v3,v4,x),euc2(v1,v2,v3,v4,y)) | -EQ(x,y). EQ(ext(x,v1,v2,v3),ext(y,v1,v2,v3)) | -EQ(x,y). EQ(ext(v1,x,v2,v3),ext(v1,y,v2,v3)) | -EQ(x,y). EQ(ext(v1,v2,x,v3),ext(v1,v2,y,v3)) | -EQ(x,y). EQ(ext(v1,v2,v3,x),ext(v1,v2,v3,y)) | -EQ(x,y). % EQ(cont(x,v1,v2,v3,v4,v5),cont(y,v1,v2,v3,v4,v5)) | -EQ(x,y). % EQ(cont(v1,x,v2,v3,v4,v5),cont(v1,y,v2,v3,v4,v5)) | -EQ(x,y). % EQ(cont(v1,v2,x,v3,v4,v5),cont(v1,v2,y,v3,v4,v5)) | -EQ(x,y). % EQ(cont(v1,v2,v3,x,v4,v5),cont(v1,v2,v3,y,v4,v5)) | -EQ(x,y). % EQ(cont(v1,v2,v3,v4,x,v5),cont(v1,v2,v3,v4,y,v5)) | -EQ(x,y). % EQ(cont(v1,v2,v3,v4,v5,x),cont(v1,v2,v3,v4,v5,y)) | -EQ(x,y). % T1 : -B(x,y,z) | B(z,y,x). % T2 : B(x,x,y). % T3 : B(x,y,y). % T6 : -B(x,z,y) | -B(x,y,z) | EQ(x,y) | EQ(y,z) | EQ(x,z). -B(y,x,z) | -B(x,y,z) | EQ(x,y) | EQ(y,z) | EQ(x,z). % T7 : -B(w,y,z) | -B(x,y,z) | B(x,w,y) | B(w,x,y) | EQ(y,z). % T8 : -B(x,y,z) | -B(x,w,z) | -B(y,v,w) | B(x,v,z). % T10 : -C(x,y,z) | C(x,z,y). -C(x,y,z) | C(y,x,z). -C(x,y,z) | C(y,z,x). -C(x,y,z) | C(z,x,y). -C(x,y,z) | C(z,y,x). % C2: -B(w,v,u) | C(u,v,w). -B(u,w,v) | C(u,v,w). -B(v,u,w) | C(u,v,w). % C3: C(x,x,y). C(x,y,x). C(y,x,x). C(x,z,y) | -EQ(x,y). % C4: -L(u,v,u1,v1) | -L(v,w,v1,w1) | -L(u,w,u1,w1) | -C(u,v,w) | C(u1,v1,w1). % C5: -C(w,v,u) | -C(x,v,u) | C(x,w,u) | EQ(u,v). -C(w,v,u) | -C(x,v,u) | C(x,w,v) | EQ(u,v). % T12: -C(u,v,w) | -C(u,v,x) | C(u,w,x) | EQ(u,v). -C(u,v,w) | -C(u,v,x) | C(v,w,x) | EQ(u,v). end_of_list. list(sos). % T11 : -C(C1,C2,C3). % denial of the theorem: -EQ(a,b). C(a,b,d1). C(a,b,d2). C(a,b,d3). -C(d1,d2,d3). end_of_list. list(demodulators). EQ(ext(x,y,z,z),y). end_of_list. weight_list(pick_and_purge). % to purge (almost all) non-ground clauses weight(x,20). weight(C1,0). weight(C2,0). weight(C3,0). weight(a,0). weight(b,0). weight(d1,0). weight(d2,0). weight(d3,0). end_of_list.