Question 24999: A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem

Original question and givens

Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).

• G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-group-homomorphism-kernel

Step-by-step derivation

source roles = [kernel-trivial-injective-boundary]

derived = [[kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary]]

source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], 0]

derived = [26, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25]], Yes]

source roles = [Kernel ORIGINAL SOURCE group, 26, FOR EVERY x,y,z in0..n-1]

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 17576, Yes]

source roles = [Kernel ORIGINAL SOURCE group, 0]

derived = [[0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 25, 0, 0], [2, 24, 0, 0], [3, 23, 0, 0], [4, 22, 0, 0], [5, 21, 0, 0], [6, 20, 0, 0], [7, 19, 0, 0], [8, 18, 0, 0], [9, 17, 0, 0], [10, 16, 0, 0], [11, 15, 0, 0], [12, 14, 0, 0], [13, 13, 0, 0], [14, 12, 0, 0], [15, 11, 0, 0], [16, 10, 0, 0], [17, 9, 0, 0], [18, 8, 0, 0], [19, 7, 0, 0], [20, 6, 0, 0], [21, 5, 0, 0], [22, 4, 0, 0], [23, 3, 0, 0], [24, 2, 0, 0], [25, 1, 0, 0]], Yes]

source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], 0]

derived = [26, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25]], Yes]

source roles = [Kernel ORIGINAL TARGET group, 26, FOR EVERY x,y,z in0..n-1]

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 17576, Yes]

source roles = [Kernel ORIGINAL TARGET group, 0]

derived = [[0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 25, 0, 0], [2, 24, 0, 0], [3, 23, 0, 0], [4, 22, 0, 0], [5, 21, 0, 0], [6, 20, 0, 0], [7, 19, 0, 0], [8, 18, 0, 0], [9, 17, 0, 0], [10, 16, 0, 0], [11, 15, 0, 0], [12, 14, 0, 0], [13, 13, 0, 0], [14, 12, 0, 0], [15, 11, 0, 0], [16, 10, 0, 0], [17, 9, 0, 0], [18, 8, 0, 0], [19, 7, 0, 0], [20, 6, 0, 0], [21, 5, 0, 0], [22, 4, 0, 0], [23, 3, 0, 0], [24, 2, 0, 0], [25, 1, 0, 0]], Yes]

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]]

derived = [[[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [0, 8, 8, 8, Yes], [0, 9, 9, 9, Yes], [0, 10, 10, 10, Yes], [0, 11, 11, 11, Yes], [0, 12, 12, 12, Yes], [0, 13, 13, 13, Yes], [0, 14, 14, 14, Yes], [0, 15, 15, 15, Yes], [0, 16, 16, 16, Yes], [0, 17, 17, 17, Yes], [0, 18, 18, 18, Yes], [0, 19, 19, 19, Yes], [0, 20, 20, 20, Yes], [0, 21, 21, 21, Yes], [0, 22, 22, 22, Yes], [0, 23, 23, 23, Yes], [0, 24, 24, 24, Yes], [0, 25, 25, 25, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 8, 8, Yes], [1, 8, 9, 9, Yes], [1, 9, 10, 10, Yes], [1, 10, 11, 11, Yes], [1, 11, 12, 12, Yes], [1, 12, 13, 13, Yes], [1, 13, 14, 14, Yes], [1, 14, 15, 15, Yes], [1, 15, 16, 16, Yes], [1, 16, 17, 17, Yes], [1, 17, 18, 18, Yes], [1, 18, 19, 19, Yes], [1, 19, 20, 20, Yes], [1, 20, 21, 21, Yes], [1, 21, 22, 22, Yes], [1, 22, 23, 23, Yes], [1, 23, 24, 24, Yes], [1, 24, 25, 25, Yes], [1, 25, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 8, 8, Yes], [2, 7, 9, 9, Yes], [2, 8, 10, 10, Yes], [2, 9, 11, 11, Yes], [2, 10, 12, 12, Yes], [2, 11, 13, 13, Yes], [2, 12, 14, 14, Yes], [2, 13, 15, 15, Yes], [2, 14, 16, 16, Yes], [2, 15, 17, 17, Yes], [2, 16, 18, 18, Yes], [2, 17, 19, 19, Yes], [2, 18, 20, 20, Yes], [2, 19, 21, 21, Yes], [2, 20, 22, 22, Yes], [2, 21, 23, 23, Yes], [2, 22, 24, 24, Yes], [2, 23, 25, 25, Yes], [2, 24, 0, 0, Yes], [2, 25, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 8, 8, Yes], [3, 6, 9, 9, Yes], [3, 7, 10, 10, Yes], [3, 8, 11, 11, Yes], [3, 9, 12, 12, Yes], [3, 10, 13, 13, Yes], [3, 11, 14, 14, Yes], [3, 12, 15, 15, Yes], [3, 13, 16, 16, Yes], [3, 14, 17, 17, Yes], [3, 15, 18, 18, Yes], [3, 16, 19, 19, Yes], [3, 17, 20, 20, Yes], [3, 18, 21, 21, Yes], [3, 19, 22, 22, Yes], [3, 20, 23, 23, Yes], [3, 21, 24, 24, Yes], [3, 22, 25, 25, Yes], [3, 23, 0, 0, Yes], [3, 24, 1, 1, Yes], [3, 25, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 8, 8, Yes], [4, 5, 9, 9, Yes], [4, 6, 10, 10, Yes], [4, 7, 11, 11, Yes], [4, 8, 12, 12, Yes], [4, 9, 13, 13, Yes], [4, 10, 14, 14, Yes], [4, 11, 15, 15, Yes], [4, 12, 16, 16, Yes], [4, 13, 17, 17, Yes], [4, 14, 18, 18, Yes], [4, 15, 19, 19, Yes], [4, 16, 20, 20, Yes], [4, 17, 21, 21, Yes], [4, 18, 22, 22, Yes], [4, 19, 23, 23, Yes], [4, 20, 24, 24, Yes], [4, 21, 25, 25, Yes], [4, 22, 0, 0, Yes], [4, 23, 1, 1, Yes], [4, 24, 2, 2, Yes], [4, 25, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 8, 8, Yes], [5, 4, 9, 9, Yes], [5, 5, 10, 10, Yes], [5, 6, 11, 11, Yes], [5, 7, 12, 12, Yes], [5, 8, 13, 13, Yes], [5, 9, 14, 14, Yes], [5, 10, 15, 15, Yes], [5, 11, 16, 16, Yes], [5, 12, 17, 17, Yes], [5, 13, 18, 18, Yes], [5, 14, 19, 19, Yes], [5, 15, 20, 20, Yes], [5, 16, 21, 21, Yes], [5, 17, 22, 22, Yes], [5, 18, 23, 23, Yes], [5, 19, 24, 24, Yes], [5, 20, 25, 25, Yes], [5, 21, 0, 0, Yes], [5, 22, 1, 1, Yes], [5, 23, 2, 2, Yes], [5, 24, 3, 3, Yes], [5, 25, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 8, 8, Yes], [6, 3, 9, 9, Yes], [6, 4, 10, 10, Yes], [6, 5, 11, 11, Yes], [6, 6, 12, 12, Yes], [6, 7, 13, 13, Yes], [6, 8, 14, 14, Yes], [6, 9, 15, 15, Yes], [6, 10, 16, 16, Yes], [6, 11, 17, 17, Yes], [6, 12, 18, 18, Yes], [6, 13, 19, 19, Yes], [6, 14, 20, 20, Yes], [6, 15, 21, 21, Yes], [6, 16, 22, 22, Yes], [6, 17, 23, 23, Yes], [6, 18, 24, 24, Yes], [6, 19, 25, 25, Yes], [6, 20, 0, 0, Yes], [6, 21, 1, 1, Yes], [6, 22, 2, 2, Yes], [6, 23, 3, 3, Yes], [6, 24, 4, 4, Yes], [6, 25, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 8, 8, Yes], [7, 2, 9, 9, Yes], [7, 3, 10, 10, Yes], [7, 4, 11, 11, Yes], [7, 5, 12, 12, Yes], [7, 6, 13, 13, Yes], [7, 7, 14, 14, Yes], [7, 8, 15, 15, Yes], [7, 9, 16, 16, Yes], [7, 10, 17, 17, Yes], [7, 11, 18, 18, Yes], [7, 12, 19, 19, Yes], [7, 13, 20, 20, Yes], [7, 14, 21, 21, Yes], [7, 15, 22, 22, Yes], [7, 16, 23, 23, Yes], [7, 17, 24, 24, Yes], [7, 18, 25, 25, Yes], [7, 19, 0, 0, Yes], [7, 20, 1, 1, Yes], [7, 21, 2, 2, Yes], [7, 22, 3, 3, Yes], [7, 23, 4, 4, Yes], [7, 24, 5, 5, Yes], [7, 25, 6, 6, Yes], [8, 0, 8, 8, Yes], [8, 1, 9, 9, Yes], [8, 2, 10, 10, Yes], [8, 3, 11, 11, Yes], [8, 4, 12, 12, Yes], [8, 5, 13, 13, Yes], [8, 6, 14, 14, Yes], [8, 7, 15, 15, Yes], [8, 8, 16, 16, Yes], [8, 9, 17, 17, Yes], [8, 10, 18, 18, Yes], [8, 11, 19, 19, Yes], [8, 12, 20, 20, Yes], [8, 13, 21, 21, Yes], [8, 14, 22, 22, Yes], [8, 15, 23, 23, Yes], [8, 16, 24, 24, Yes], [8, 17, 25, 25, Yes], [8, 18, 0, 0, Yes], [8, 19, 1, 1, Yes], [8, 20, 2, 2, Yes], [8, 21, 3, 3, Yes], [8, 22, 4, 4, Yes], [8, 23, 5, 5, Yes], [8, 24, 6, 6, Yes], [8, 25, 7, 7, Yes], [9, 0, 9, 9, Yes], [9, 1, 10, 10, Yes], [9, 2, 11, 11, Yes], [9, 3, 12, 12, Yes], [9, 4, 13, 13, Yes], [9, 5, 14, 14, Yes], [9, 6, 15, 15, Yes], [9, 7, 16, 16, Yes], [9, 8, 17, 17, Yes], [9, 9, 18, 18, Yes], [9, 10, 19, 19, Yes], [9, 11, 20, 20, Yes], [9, 12, 21, 21, Yes], [9, 13, 22, 22, Yes], [9, 14, 23, 23, Yes], [9, 15, 24, 24, Yes], [9, 16, 25, 25, Yes], [9, 17, 0, 0, Yes], [9, 18, 1, 1, Yes], [9, 19, 2, 2, Yes], [9, 20, 3, 3, Yes], [9, 21, 4, 4, Yes], [9, 22, 5, 5, Yes], [9, 23, 6, 6, Yes], [9, 24, 7, 7, Yes], [9, 25, 8, 8, Yes], [10, 0, 10, 10, Yes], [10, 1, 11, 11, Yes], [10, 2, 12, 12, Yes], [10, 3, 13, 13, Yes], [10, 4, 14, 14, Yes], [10, 5, 15, 15, Yes], [10, 6, 16, 16, Yes], [10, 7, 17, 17, Yes], [10, 8, 18, 18, Yes], [10, 9, 19, 19, Yes], [10, 10, 20, 20, Yes], [10, 11, 21, 21, Yes], [10, 12, 22, 22, Yes], [10, 13, 23, 23, Yes], [10, 14, 24, 24, Yes], [10, 15, 25, 25, Yes], [10, 16, 0, 0, Yes], [10, 17, 1, 1, Yes], [10, 18, 2, 2, Yes], [10, 19, 3, 3, Yes], [10, 20, 4, 4, Yes], [10, 21, 5, 5, Yes], [10, 22, 6, 6, Yes], [10, 23, 7, 7, Yes], [10, 24, 8, 8, Yes], [10, 25, 9, 9, Yes], [11, 0, 11, 11, Yes], [11, 1, 12, 12, Yes], [11, 2, 13, 13, Yes], [11, 3, 14, 14, Yes], [11, 4, 15, 15, Yes], [11, 5, 16, 16, Yes], [11, 6, 17, 17, Yes], [11, 7, 18, 18, Yes], [11, 8, 19, 19, Yes], [11, 9, 20, 20, Yes], [11, 10, 21, 21, Yes], [11, 11, 22, 22, Yes], [11, 12, 23, 23, Yes], [11, 13, 24, 24, Yes], [11, 14, 25, 25, Yes], [11, 15, 0, 0, Yes], [11, 16, 1, 1, Yes], [11, 17, 2, 2, Yes], [11, 18, 3, 3, Yes], [11, 19, 4, 4, Yes], [11, 20, 5, 5, Yes], [11, 21, 6, 6, Yes], [11, 22, 7, 7, Yes], [11, 23, 8, 8, Yes], [11, 24, 9, 9, Yes], [11, 25, 10, 10, Yes], [12, 0, 12, 12, Yes], [12, 1, 13, 13, Yes], [12, 2, 14, 14, Yes], [12, 3, 15, 15, Yes], [12, 4, 16, 16, Yes], [12, 5, 17, 17, Yes], [12, 6, 18, 18, Yes], [12, 7, 19, 19, Yes], [12, 8, 20, 20, Yes], [12, 9, 21, 21, Yes], [12, 10, 22, 22, Yes], [12, 11, 23, 23, Yes], [12, 12, 24, 24, Yes], [12, 13, 25, 25, Yes], [12, 14, 0, 0, Yes], [12, 15, 1, 1, Yes], [12, 16, 2, 2, Yes], [12, 17, 3, 3, Yes], [12, 18, 4, 4, Yes], [12, 19, 5, 5, Yes], [12, 20, 6, 6, Yes], [12, 21, 7, 7, Yes], [12, 22, 8, 8, Yes], [12, 23, 9, 9, Yes], [12, 24, 10, 10, Yes], [12, 25, 11, 11, Yes], [13, 0, 13, 13, Yes], [13, 1, 14, 14, Yes], [13, 2, 15, 15, Yes], [13, 3, 16, 16, Yes], [13, 4, 17, 17, Yes], [13, 5, 18, 18, Yes], [13, 6, 19, 19, Yes], [13, 7, 20, 20, Yes], [13, 8, 21, 21, Yes], [13, 9, 22, 22, Yes], [13, 10, 23, 23, Yes], [13, 11, 24, 24, Yes], [13, 12, 25, 25, Yes], [13, 13, 0, 0, Yes], [13, 14, 1, 1, Yes], [13, 15, 2, 2, Yes], [13, 16, 3, 3, Yes], [13, 17, 4, 4, Yes], [13, 18, 5, 5, Yes], [13, 19, 6, 6, Yes], [13, 20, 7, 7, Yes], [13, 21, 8, 8, Yes], [13, 22, 9, 9, Yes], [13, 23, 10, 10, Yes], [13, 24, 11, 11, Yes], [13, 25, 12, 12, Yes], [14, 0, 14, 14, Yes], [14, 1, 15, 15, Yes], [14, 2, 16, 16, Yes], [14, 3, 17, 17, Yes], [14, 4, 18, 18, Yes], [14, 5, 19, 19, Yes], [14, 6, 20, 20, Yes], [14, 7, 21, 21, Yes], [14, 8, 22, 22, Yes], [14, 9, 23, 23, Yes], [14, 10, 24, 24, Yes], [14, 11, 25, 25, Yes], [14, 12, 0, 0, Yes], [14, 13, 1, 1, Yes], [14, 14, 2, 2, Yes], [14, 15, 3, 3, Yes], [14, 16, 4, 4, Yes], [14, 17, 5, 5, Yes], [14, 18, 6, 6, Yes], [14, 19, 7, 7, Yes], [14, 20, 8, 8, Yes], [14, 21, 9, 9, Yes], [14, 22, 10, 10, Yes], [14, 23, 11, 11, Yes], [14, 24, 12, 12, Yes], [14, 25, 13, 13, Yes], [15, 0, 15, 15, Yes], [15, 1, 16, 16, Yes], [15, 2, 17, 17, Yes], [15, 3, 18, 18, Yes], [15, 4, 19, 19, Yes], [15, 5, 20, 20, Yes], [15, 6, 21, 21, Yes], [15, 7, 22, 22, Yes], [15, 8, 23, 23, Yes], [15, 9, 24, 24, Yes], [15, 10, 25, 25, Yes], [15, 11, 0, 0, Yes], [15, 12, 1, 1, Yes], [15, 13, 2, 2, Yes], [15, 14, 3, 3, Yes], [15, 15, 4, 4, Yes], [15, 16, 5, 5, Yes], [15, 17, 6, 6, Yes], [15, 18, 7, 7, Yes], [15, 19, 8, 8, Yes], [15, 20, 9, 9, Yes], [15, 21, 10, 10, Yes], [15, 22, 11, 11, Yes], [15, 23, 12, 12, Yes], [15, 24, 13, 13, Yes], [15, 25, 14, 14, Yes], [16, 0, 16, 16, Yes], [16, 1, 17, 17, Yes], [16, 2, 18, 18, Yes], [16, 3, 19, 19, Yes], [16, 4, 20, 20, Yes], [16, 5, 21, 21, Yes], [16, 6, 22, 22, Yes], [16, 7, 23, 23, Yes], [16, 8, 24, 24, Yes], [16, 9, 25, 25, Yes], [16, 10, 0, 0, Yes], [16, 11, 1, 1, Yes], [16, 12, 2, 2, Yes], [16, 13, 3, 3, Yes], [16, 14, 4, 4, Yes], [16, 15, 5, 5, Yes], [16, 16, 6, 6, Yes], [16, 17, 7, 7, Yes], [16, 18, 8, 8, Yes], [16, 19, 9, 9, Yes], [16, 20, 10, 10, Yes], [16, 21, 11, 11, Yes], [16, 22, 12, 12, Yes], [16, 23, 13, 13, Yes], [16, 24, 14, 14, Yes], [16, 25, 15, 15, Yes], [17, 0, 17, 17, Yes], [17, 1, 18, 18, Yes], [17, 2, 19, 19, Yes], [17, 3, 20, 20, Yes], [17, 4, 21, 21, Yes], [17, 5, 22, 22, Yes], [17, 6, 23, 23, Yes], [17, 7, 24, 24, Yes], [17, 8, 25, 25, Yes], [17, 9, 0, 0, Yes], [17, 10, 1, 1, Yes], [17, 11, 2, 2, Yes], [17, 12, 3, 3, Yes], [17, 13, 4, 4, Yes], [17, 14, 5, 5, Yes], [17, 15, 6, 6, Yes], [17, 16, 7, 7, Yes], [17, 17, 8, 8, Yes], [17, 18, 9, 9, Yes], [17, 19, 10, 10, Yes], [17, 20, 11, 11, Yes], [17, 21, 12, 12, Yes], [17, 22, 13, 13, Yes], [17, 23, 14, 14, Yes], [17, 24, 15, 15, Yes], [17, 25, 16, 16, Yes], [18, 0, 18, 18, Yes], [18, 1, 19, 19, Yes], [18, 2, 20, 20, Yes], [18, 3, 21, 21, Yes], [18, 4, 22, 22, Yes], [18, 5, 23, 23, Yes], [18, 6, 24, 24, Yes], [18, 7, 25, 25, Yes], [18, 8, 0, 0, Yes], [18, 9, 1, 1, Yes], [18, 10, 2, 2, Yes], [18, 11, 3, 3, Yes], [18, 12, 4, 4, Yes], [18, 13, 5, 5, Yes], [18, 14, 6, 6, Yes], [18, 15, 7, 7, Yes], [18, 16, 8, 8, Yes], [18, 17, 9, 9, Yes], [18, 18, 10, 10, Yes], [18, 19, 11, 11, Yes], [18, 20, 12, 12, Yes], [18, 21, 13, 13, Yes], [18, 22, 14, 14, Yes], [18, 23, 15, 15, Yes], [18, 24, 16, 16, Yes], [18, 25, 17, 17, Yes], [19, 0, 19, 19, Yes], [19, 1, 20, 20, Yes], [19, 2, 21, 21, Yes], [19, 3, 22, 22, Yes], [19, 4, 23, 23, Yes], [19, 5, 24, 24, Yes], [19, 6, 25, 25, Yes], [19, 7, 0, 0, Yes], [19, 8, 1, 1, Yes], [19, 9, 2, 2, Yes], [19, 10, 3, 3, Yes], [19, 11, 4, 4, Yes], [19, 12, 5, 5, Yes], [19, 13, 6, 6, Yes], [19, 14, 7, 7, Yes], [19, 15, 8, 8, Yes], [19, 16, 9, 9, Yes], [19, 17, 10, 10, Yes], [19, 18, 11, 11, Yes], [19, 19, 12, 12, Yes], [19, 20, 13, 13, Yes], [19, 21, 14, 14, Yes], [19, 22, 15, 15, Yes], [19, 23, 16, 16, Yes], [19, 24, 17, 17, Yes], [19, 25, 18, 18, Yes], [20, 0, 20, 20, Yes], [20, 1, 21, 21, Yes], [20, 2, 22, 22, Yes], [20, 3, 23, 23, Yes], [20, 4, 24, 24, Yes], [20, 5, 25, 25, Yes], [20, 6, 0, 0, Yes], [20, 7, 1, 1, Yes], [20, 8, 2, 2, Yes], [20, 9, 3, 3, Yes], [20, 10, 4, 4, Yes], [20, 11, 5, 5, Yes], [20, 12, 6, 6, Yes], [20, 13, 7, 7, Yes], [20, 14, 8, 8, Yes], [20, 15, 9, 9, Yes], [20, 16, 10, 10, Yes], [20, 17, 11, 11, Yes], [20, 18, 12, 12, Yes], [20, 19, 13, 13, Yes], [20, 20, 14, 14, Yes], [20, 21, 15, 15, Yes], [20, 22, 16, 16, Yes], [20, 23, 17, 17, Yes], [20, 24, 18, 18, Yes], [20, 25, 19, 19, Yes], [21, 0, 21, 21, Yes], [21, 1, 22, 22, Yes], [21, 2, 23, 23, Yes], [21, 3, 24, 24, Yes], [21, 4, 25, 25, Yes], [21, 5, 0, 0, Yes], [21, 6, 1, 1, Yes], [21, 7, 2, 2, Yes], [21, 8, 3, 3, Yes], [21, 9, 4, 4, Yes], [21, 10, 5, 5, Yes], [21, 11, 6, 6, Yes], [21, 12, 7, 7, Yes], [21, 13, 8, 8, Yes], [21, 14, 9, 9, Yes], [21, 15, 10, 10, Yes], [21, 16, 11, 11, Yes], [21, 17, 12, 12, Yes], [21, 18, 13, 13, Yes], [21, 19, 14, 14, Yes], [21, 20, 15, 15, Yes], [21, 21, 16, 16, Yes], [21, 22, 17, 17, Yes], [21, 23, 18, 18, Yes], [21, 24, 19, 19, Yes], [21, 25, 20, 20, Yes], [22, 0, 22, 22, Yes], [22, 1, 23, 23, Yes], [22, 2, 24, 24, Yes], [22, 3, 25, 25, Yes], [22, 4, 0, 0, Yes], [22, 5, 1, 1, Yes], [22, 6, 2, 2, Yes], [22, 7, 3, 3, Yes], [22, 8, 4, 4, Yes], [22, 9, 5, 5, Yes], [22, 10, 6, 6, Yes], [22, 11, 7, 7, Yes], [22, 12, 8, 8, Yes], [22, 13, 9, 9, Yes], [22, 14, 10, 10, Yes], [22, 15, 11, 11, Yes], [22, 16, 12, 12, Yes], [22, 17, 13, 13, Yes], [22, 18, 14, 14, Yes], [22, 19, 15, 15, Yes], [22, 20, 16, 16, Yes], [22, 21, 17, 17, Yes], [22, 22, 18, 18, Yes], [22, 23, 19, 19, Yes], [22, 24, 20, 20, Yes], [22, 25, 21, 21, Yes], [23, 0, 23, 23, Yes], [23, 1, 24, 24, Yes], [23, 2, 25, 25, Yes], [23, 3, 0, 0, Yes], [23, 4, 1, 1, Yes], [23, 5, 2, 2, Yes], [23, 6, 3, 3, Yes], [23, 7, 4, 4, Yes], [23, 8, 5, 5, Yes], [23, 9, 6, 6, Yes], [23, 10, 7, 7, Yes], [23, 11, 8, 8, Yes], [23, 12, 9, 9, Yes], [23, 13, 10, 10, Yes], [23, 14, 11, 11, Yes], [23, 15, 12, 12, Yes], [23, 16, 13, 13, Yes], [23, 17, 14, 14, Yes], [23, 18, 15, 15, Yes], [23, 19, 16, 16, Yes], [23, 20, 17, 17, Yes], [23, 21, 18, 18, Yes], [23, 22, 19, 19, Yes], [23, 23, 20, 20, Yes], [23, 24, 21, 21, Yes], [23, 25, 22, 22, Yes], [24, 0, 24, 24, Yes], [24, 1, 25, 25, Yes], [24, 2, 0, 0, Yes], [24, 3, 1, 1, Yes], [24, 4, 2, 2, Yes], [24, 5, 3, 3, Yes], [24, 6, 4, 4, Yes], [24, 7, 5, 5, Yes], [24, 8, 6, 6, Yes], [24, 9, 7, 7, Yes], [24, 10, 8, 8, Yes], [24, 11, 9, 9, Yes], [24, 12, 10, 10, Yes], [24, 13, 11, 11, Yes], [24, 14, 12, 12, Yes], [24, 15, 13, 13, Yes], [24, 16, 14, 14, Yes], [24, 17, 15, 15, Yes], [24, 18, 16, 16, Yes], [24, 19, 17, 17, Yes], [24, 20, 18, 18, Yes], [24, 21, 19, 19, Yes], [24, 22, 20, 20, Yes], [24, 23, 21, 21, Yes], [24, 24, 22, 22, Yes], [24, 25, 23, 23, Yes], [25, 0, 25, 25, Yes], [25, 1, 0, 0, Yes], [25, 2, 1, 1, Yes], [25, 3, 2, 2, Yes], [25, 4, 3, 3, Yes], [25, 5, 4, 4, Yes], [25, 6, 5, 5, Yes], [25, 7, 6, 6, Yes], [25, 8, 7, 7, Yes], [25, 9, 8, 8, Yes], [25, 10, 9, 9, Yes], [25, 11, 10, 10, Yes], [25, 12, 11, 11, Yes], [25, 13, 12, 12, Yes], [25, 14, 13, 13, Yes], [25, 15, 14, 14, Yes], [25, 16, 15, 15, Yes], [25, 17, 16, 16, Yes], [25, 18, 17, 17, Yes], [25, 19, 18, 18, Yes], [25, 20, 19, 19, Yes], [25, 21, 20, 20, Yes], [25, 22, 21, 21, Yes], [25, 23, 22, 22, Yes], [25, 24, 23, 23, Yes], [25, 25, 24, 24, Yes]], Yes, null]

source roles = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], 0, 0]

derived = [[0], 0, 0]

source roles = [Kernel ACTUAL identity preimage, [0], 0]

derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes]

source roles = [Kernel ACTUAL identity preimage, 0, [0], 0]

derived = [[[0, 0, 0, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 1, [0], 25]

derived = [[[0, 1, 25, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 2, [0], 24]

derived = [[[0, 2, 24, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 3, [0], 23]

derived = [[[0, 3, 23, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 4, [0], 22]

derived = [[[0, 4, 22, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 5, [0], 21]

derived = [[[0, 5, 21, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 6, [0], 20]

derived = [[[0, 6, 20, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 7, [0], 19]

derived = [[[0, 7, 19, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 8, [0], 18]

derived = [[[0, 8, 18, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 9, [0], 17]

derived = [[[0, 9, 17, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 10, [0], 16]

derived = [[[0, 10, 16, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 11, [0], 15]

derived = [[[0, 11, 15, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 12, [0], 14]

derived = [[[0, 12, 14, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 13, [0], 13]

derived = [[[0, 13, 13, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 14, [0], 12]

derived = [[[0, 14, 12, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 15, [0], 11]

derived = [[[0, 15, 11, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 16, [0], 10]

derived = [[[0, 16, 10, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 17, [0], 9]

derived = [[[0, 17, 9, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 18, [0], 8]

derived = [[[0, 18, 8, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 19, [0], 7]

derived = [[[0, 19, 7, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 20, [0], 6]

derived = [[[0, 20, 6, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 21, [0], 5]

derived = [[[0, 21, 5, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 22, [0], 4]

derived = [[[0, 22, 4, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 23, [0], 3]

derived = [[[0, 23, 3, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 24, [0], 2]

derived = [[[0, 24, 2, 0]], [0], Yes]

source roles = [Kernel ACTUAL identity preimage, 25, [0], 1]

derived = [[[0, 25, 1, 0]], [0], Yes]

source roles = [[0], Yes, 0]

derived = [f(kl)=f(k)f(l)=e, f(k^-1)=f(k)^-1=e, f(gkg^-1)=f(g)e*f(g)^-1=e]

source roles = [0, [0], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]]

derived = [[[0, [0]], [1, [1]], [2, [2]], [3, [3]], [4, [4]], [5, [5]], [6, [6]], [7, [7]], [8, [8]], [9, [9]], [10, [10]], [11, [11]], [12, [12]], [13, [13]], [14, [14]], [15, [15]], [16, [16]], [17, [17]], [18, [18]], [19, [19]], [20, [20]], [21, [21]], [22, [22]], [23, [23]], [24, [24]], [25, [25]]], Yes, Yes, Yes, No]

Reverse checks

• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.