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needsPackage "Polyhedra" assert( (M= matrix{{1,0,0},{0,1,0},{1,0,1},{0,1,1}}) === map((ZZ)^4,(ZZ)^3,{{1, 0, 0}, {0, 1, 0}, {1, 0, 1}, {0, 1, 1}}) ); C=posHull transpose M assert( (rays C) === map((ZZ)^3,(ZZ)^4,{{1, 0, 1, 0}, {0, 1, 0, 1}, {0, 0, 1, 1}}) ); assert( (fVector C) === {1,4,4,1} ); Cv = dualCone C assert( (rays Cv) === map((ZZ)^3,(ZZ)^4,{{1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, -1}}) ); assert( (set hilbertBasis C) === new Set from {map((ZZ)^3,(ZZ)^1,{{1}, {0}, {0}}),map((ZZ)^3,(ZZ)^1,{{0}, {1}, {0}}),map((ZZ)^3,(ZZ)^1,{{0}, {1}, {1}}),map((ZZ)^3,(ZZ)^1,{{1}, {0}, {1}})} ); C1 = posHull transpose matrix {{0,-1,0},{0,0,1},{ -1,0,0}}; assert( (rays C1) === map((ZZ)^3,(ZZ)^3,{{-1, 0, 0}, {0, -1, 0}, {0, 0, 1}}) ); assert( (linealitySpace C1) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C1) === 3 ); assert( (isPointed C1) === true ); assert( (isSimplicial C1) === true ); C2 = posHull transpose matrix {{0,-1,0},{0,0,-1},{ -1,0,0}}; assert( (rays C2) === map((ZZ)^3,(ZZ)^3,{{-1, 0, 0}, {0, -1, 0}, {0, 0, -1}}) ); assert( (linealitySpace C2) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C2) === 3 ); assert( (isPointed C2) === true ); assert( (isSimplicial C2) === true ); C3 = posHull transpose matrix {{0,-1,0},{0,0,1},{1,0,0}}; assert( (rays C3) === map((ZZ)^3,(ZZ)^3,{{1, 0, 0}, {0, -1, 0}, {0, 0, 1}}) ); assert( (linealitySpace C3) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C3) === 3 ); assert( (isPointed C3) === true ); assert( (isSimplicial C3) === true ); C4 = posHull transpose matrix {{0,-1,0},{0,0,-1},{1,0,0}}; assert( (rays C4) === map((ZZ)^3,(ZZ)^3,{{1, 0, 0}, {0, -1, 0}, {0, 0, -1}}) ); assert( (linealitySpace C4) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C4) === 3 ); assert( (isPointed C4) === true ); assert( (isSimplicial C4) === true ); C5 = posHull transpose matrix {{0,1,0},{0,0,1},{ -1,0,0}}; assert( (rays C5) === map((ZZ)^3,(ZZ)^3,{{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}}) ); assert( (linealitySpace C5) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C5) === 3 ); assert( (isPointed C5) === true ); assert( (isSimplicial C5) === true ); C6 = posHull transpose matrix {{0,1,0},{0,0,-1},{ -1,0,0}}; assert( (rays C6) === map((ZZ)^3,(ZZ)^3,{{-1, 0, 0}, {0, 1, 0}, {0, 0, -1}}) ); assert( (linealitySpace C6) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C6) === 3 ); assert( (isPointed C6) === true ); assert( (isSimplicial C6) === true ); C7 = posHull transpose matrix {{0,1,0},{0,0,-1},{1,0,0}}; assert( (rays C7) === map((ZZ)^3,(ZZ)^3,{{1, 0, 0}, {0, 1, 0}, {0, 0, -1}}) ); assert( (linealitySpace C7) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim C7) === 3 ); assert( (isPointed C7) === true ); assert( (isSimplicial C7) === true ); B1 = posHull transpose matrix {{1,0,0},{0,0,1},{1,1,2}}; assert( (rays B1) === map((ZZ)^3,(ZZ)^3,{{1, 0, 1}, {0, 0, 1}, {0, 1, 2}}) ); assert( (linealitySpace B1) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B1) === 3 ); assert( (isPointed B1) === true ); assert( (isSimplicial B1) === true ); B2 = posHull transpose matrix {{1,0,0},{0,1,0},{2,1,1}}; assert( (rays B2) === map((ZZ)^3,(ZZ)^3,{{1, 0, 2}, {0, 1, 1}, {0, 0, 1}}) ); assert( (linealitySpace B2) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B2) === 3 ); assert( (isPointed B2) === true ); assert( (isSimplicial B2) === true ); B3 = posHull transpose matrix {{0,0,1},{0,1,0},{1,2,1}}; assert( (rays B3) === map((ZZ)^3,(ZZ)^3,{{0, 0, 1}, {1, 0, 2}, {0, 1, 1}}) ); assert( (linealitySpace B3) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B3) === 3 ); assert( (isPointed B3) === true ); assert( (isSimplicial B3) === true ); B4 = posHull transpose matrix {{0,0,1},{1,1,2},{1,2,1}}; assert( (rays B4) === map((ZZ)^3,(ZZ)^3,{{0, 1, 1}, {0, 2, 1}, {1, 1, 2}}) ); assert( (linealitySpace B4) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B4) === 3 ); assert( (isPointed B4) === true ); assert( (isSimplicial B4) === true ); B5 = posHull transpose matrix {{0,1,0},{2,1,1},{1,2,1}}; assert( (rays B5) === map((ZZ)^3,(ZZ)^3,{{0, 2, 1}, {1, 1, 2}, {0, 1, 1}}) ); assert( (linealitySpace B5) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B5) === 3 ); assert( (isPointed B5) === true ); assert( (isSimplicial B5) === true ); B6 = posHull transpose matrix {{1,0,0},{2,1,1},{1,1,2}}; assert( (rays B6) === map((ZZ)^3,(ZZ)^3,{{1, 2, 1}, {0, 1, 1}, {0, 1, 2}}) ); assert( (linealitySpace B6) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B6) === 3 ); assert( (isPointed B6) === true ); assert( (isSimplicial B6) === true ); B7 = posHull transpose matrix {{1,1,1},{2,1,1},{1,2,1}}; assert( (rays B7) === map((ZZ)^3,(ZZ)^3,{{1, 2, 1}, {1, 1, 2}, {1, 1, 1}}) ); assert( (linealitySpace B7) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B7) === 3 ); assert( (isPointed B7) === true ); assert( (isSimplicial B7) === true ); B8 = posHull transpose matrix {{1,1,1},{1,1,2},{1,2,1}}; assert( (rays B8) === map((ZZ)^3,(ZZ)^3,{{1, 1, 1}, {1, 2, 1}, {1, 1, 2}}) ); assert( (linealitySpace B8) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B8) === 3 ); assert( (isPointed B8) === true ); assert( (isSimplicial B8) === true ); B9 = posHull transpose matrix {{1,1,1},{1,1,2},{2,1,1}}; assert( (rays B9) === map((ZZ)^3,(ZZ)^3,{{1, 2, 1}, {1, 1, 1}, {1, 1, 2}}) ); assert( (linealitySpace B9) === map((ZZ)^3,(ZZ)^0,0) ); assert( (dim B9) === 3 ); assert( (isPointed B9) === true ); assert( (isSimplicial B9) === true ); F=fan{C1,C2,C3,C4,C5,C6,C7,B1,B2,B3,B4,B5,B6,B7,B8,B9} assert( (isPolytopal F) === false ); end print generateAssertions /// needsPackage "Polyhedra" M= matrix{{1,0,0},{0,1,0},{1,0,1},{0,1,1}} C=posHull transpose M rays C fVector C Cv = dualCone C rays Cv set hilbertBasis C C1 = posHull transpose matrix {{0,-1,0},{0,0,1},{ -1,0,0}}; rays C1 linealitySpace C1 dim C1 isPointed C1 isSimplicial C1 C2 = posHull transpose matrix {{0,-1,0},{0,0,-1},{ -1,0,0}}; rays C2 linealitySpace C2 dim C2 isPointed C2 isSimplicial C2 C3 = posHull transpose matrix {{0,-1,0},{0,0,1},{1,0,0}}; rays C3 linealitySpace C3 dim C3 isPointed C3 isSimplicial C3 C4 = posHull transpose matrix {{0,-1,0},{0,0,-1},{1,0,0}}; rays C4 linealitySpace C4 dim C4 isPointed C4 isSimplicial C4 C5 = posHull transpose matrix {{0,1,0},{0,0,1},{ -1,0,0}}; rays C5 linealitySpace C5 dim C5 isPointed C5 isSimplicial C5 C6 = posHull transpose matrix {{0,1,0},{0,0,-1},{ -1,0,0}}; rays C6 linealitySpace C6 dim C6 isPointed C6 isSimplicial C6 C7 = posHull transpose matrix {{0,1,0},{0,0,-1},{1,0,0}}; rays C7 linealitySpace C7 dim C7 isPointed C7 isSimplicial C7 B1 = posHull transpose matrix {{1,0,0},{0,0,1},{1,1,2}}; rays B1 linealitySpace B1 dim B1 isPointed B1 isSimplicial B1 B2 = posHull transpose matrix {{1,0,0},{0,1,0},{2,1,1}}; rays B2 linealitySpace B2 dim B2 isPointed B2 isSimplicial B2 B3 = posHull transpose matrix {{0,0,1},{0,1,0},{1,2,1}}; rays B3 linealitySpace B3 dim B3 isPointed B3 isSimplicial B3 B4 = posHull transpose matrix {{0,0,1},{1,1,2},{1,2,1}}; rays B4 linealitySpace B4 dim B4 isPointed B4 isSimplicial B4 B5 = posHull transpose matrix {{0,1,0},{2,1,1},{1,2,1}}; rays B5 linealitySpace B5 dim B5 isPointed B5 isSimplicial B5 B6 = posHull transpose matrix {{1,0,0},{2,1,1},{1,1,2}}; rays B6 linealitySpace B6 dim B6 isPointed B6 isSimplicial B6 B7 = posHull transpose matrix {{1,1,1},{2,1,1},{1,2,1}}; rays B7 linealitySpace B7 dim B7 isPointed B7 isSimplicial B7 B8 = posHull transpose matrix {{1,1,1},{1,1,2},{1,2,1}}; rays B8 linealitySpace B8 dim B8 isPointed B8 isSimplicial B8 B9 = posHull transpose matrix {{1,1,1},{1,1,2},{2,1,1}}; rays B9 linealitySpace B9 dim B9 isPointed B9 isSimplicial B9 F=fan{C1,C2,C3,C4,C5,C6,C7,B1,B2,B3,B4,B5,B6,B7,B8,B9} isPolytopal F ///