Theorem: ############################# Let , f[1], be the value of , [T $ a[1], B, T $ a[2], B, T $ a[3], F, T $ a[4], F] ############################# Then we have: f[1](0, 0, 0, 0) = [-4] f[1](1, 0, 0, 0) = [-1] f[1](2, 0, 0, 0) = [{[0]}, {[0]}] f[1](a[1], 0, 0, 0) = [{[a[1] - 5/2]}, {[0]}], 3 <= a[1] f[1](0, 1, 0, 0) = [-3/2] f[1](1, 1, 0, 0) = [0] f[1](2, 1, 0, 0) = [{[1]}, {[{[1/2]}, {[0]}]}] f[1](a[1], 1, 0, 0) = [a[1] - 3/2], 3 <= a[1] f[1](0, 2, 0, 0) = [{[{[0]}, {[{[0]}, {[0]}]}]}, {[-1/4]}] f[1](1, 2, 0, 0) = [{[1/2]}, {[1/2]}] f[1](a[1], 2, 0, 0) = [a[1] - 1], 2 <= a[1] f[1](a[1], a[2], 0, 0) = [{[a[1]]}, {[a[1]]}], 0 <= a[1], 3 <= a[2] f[1](0, 0, 1, 0) = [-2] f[1](1, 0, 1, 0) = [{[0]}, {[0]}] f[1](2, 0, 1, 0) = [1] f[1](3, 0, 1, 0) = [1] f[1](a[1], 0, 1, 0) = [{[{[a[1] - 2]}, {[a[1] - 2]}]}, {[{[1]}, {[1]}]}], 4 <= a[1] f[1](0, 1, 1, 0) = [{[0]}, {[{[0]}, {[0]}]}] f[1](1, 1, 1, 0) = [{[2]}, {[{[1/2]}, {[1/2]}]}] f[1](a[1], 1, 1, 0) = [{[2 a[1]]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}], 2 <= a[1] f[1](a[1], a[2], 1, 0) = [{[2 a[1] + a[2] - 1]}, {[{[a[1]]}, {[a[1]]}]}], 0 <= a[1], 2 <= a[2] f[1](a[1], a[2], a[3], a[4]) = [2 a[1] + a[2]], 0 <= a[1], 0 <= a[2], 2 <= a[3], 0 <= a[4] f[1](0, 0, 0, 1) = [-3] f[1](a[1], 0, 0, 1) = [{[a[1] - 2]}, {[1]}], 1 <= a[1] f[1](0, 1, 0, 1) = [-1/2] f[1](a[1], 1, 0, 1) = [{[a[1] - 1/2]}, {[{[a[1] - 1]}, {[1]}]}], 1 <= a[1] f[1](a[1], 2, 0, 1) = [{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}], 0 <= a[1] f[1](a[1], a[2], 0, 1) = [{[{[2 a[1] + a[2] - 2]}, {[a[1] + 1/2]}]}, {[a[1]]}], 0 <= a[1], 3 <= a[2] f[1](0, 0, 1, 1) = [-1] f[1](a[1], 0, 1, 1) = [a[1] - 1/2], 1 <= a[1] f[1](a[1], 1, 1, 1) = [{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}], 0 <= a[1] f[1](a[1], a[2], 1, 1) = [{[2 a[1] + a[2] - 1]}, {[a[1] + 1/2]}], 0 <= a[1], 2 <= a[2] f[1](a[1], a[2], a[3], a[4]) = [2 a[1] + a[2]], 0 <= a[1], 0 <= a[2], 2 <= a[3], 0 <= a[4] f[1](0, 0, 0, a[4]) = [-2], 2 <= a[4] f[1](a[1], 0, 0, a[4]) = [a[1] - 1], 1 <= a[1], 2 <= a[4] f[1](0, 1, 0, a[4]) = [-1/2], 2 <= a[4] f[1](a[1], 1, 0, a[4]) = [{[a[1]]}, {[a[1]]}], 1 <= a[1], 2 <= a[4] f[1](a[1], a[2], 0, a[4]) = [{[{[2 a[1] + a[2] - 2]}, {[1 + a[1]]}]}, {[a[1]]}] , 0 <= a[1], 2 <= a[2], 2 <= a[4] f[1](0, 0, 1, a[4]) = [-1], 2 <= a[4] f[1](a[1], 0, 1, a[4]) = [a[1]], 1 <= a[1], 2 <= a[4] f[1](a[1], a[2], 1, a[4]) = [{[2 a[1] + a[2] - 1]}, {[1 + a[1]]}], 0 <= a[1], 2 <= a[4], 1 <= a[2] Theorem: ############################# Let , f[2], be the value of , [T $ a[1], B, T $ a[2], F, T $ a[3], B, T $ a[4], F] ############################# Then we have: f[2](0, 0, 0, 0) = [-3] f[2](0, 0, 1, 0) = [-3/2] f[2](0, 0, a[3], 0) = [{[{[a[3] - 3]}, {[0]}]}, {[-1]}], 2 <= a[3] f[2](0, 0, 0, 1) = [-2] f[2](0, 0, a[3], 1) = [{[a[3] - 2]}, {[0]}], 1 <= a[3] f[2](0, 0, a[3], a[4]) = [a[3] - 1], 2 <= a[4], 0 <= a[3] f[2](1, 0, 0, 0) = [{[-1]}, {[-1]}] f[2](a[1], 0, 0, 0) = [0], 2 <= a[1] f[2](0, 1, 0, 0) = [-1] f[2](1, 1, 0, 0) = [{[0]}, {[0]}] f[2](a[1], 1, 0, 0) = [{[a[1] - 3/2]}, {[0]}], 2 <= a[1] f[2](0, 2, 0, 0) = [-1/4] f[2](a[1], 2, 0, 0) = [a[1] - 1/2], 1 <= a[1] f[2](a[1], 3, 0, 0) = [{[{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}]}, {[a[1]]}] , 0 <= a[1] f[2](a[1], a[2], 0, 0) = [{[{[{[2 a[1] + a[2] - 3]}, {[a[1] + 1/2]}]}, {[a[1]]}]}, {[a[1]]}], 0 <= a[1], 4 <= a[2] f[2](1, 0, 1, 0) = [0] f[2](2, 0, 1, 0) = [{[1]}, {[{[1/2]}, {[0]}]}] f[2](a[1], 0, 1, 0) = [{[1]}, {[1]}], 3 <= a[1] f[2](a[1], 0, a[3], 0) = [ {[{[a[1] + a[3] - 5/2]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}]}, {[{[{[a[1] - 2]}, {[1]}]}, {[0]}]}], 2 <= a[3], 1 <= a[1] f[2](0, 1, 1, 0) = [{[0]}, {[0]}] f[2](1, 1, 1, 0) = [{[1/2]}, {[1/2]}] f[2](a[1], 1, 1, 0) = [{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}], 2 <= a[1] f[2](a[1], 1, a[3], 0) = [ {[{[a[1] + a[3] - 2]}, {[{[a[1]]}, {[2]}]}]}, {[2], [{[a[1] - 1]}, {[1]}]}] , 0 <= a[1], 2 <= a[3] f[2](a[1], a[2], a[3], 0) = [{[{[a[1] + a[3] - 2]}, {[1 + a[1]]}]}, {[a[1]]}], 0 <= a[1], 2 <= a[2], 1 <= a[3] f[2](1, 0, 0, 1) = [{[0]}, {[0]}] f[2](a[1], 0, 0, 1) = [1], 2 <= a[1] f[2](a[1], 0, a[3], 1) = [{[a[1] - 3/2 + a[3]]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}], 1 <= a[1], 1 <= a[3] f[2](a[1], 1, 0, 1) = [{[a[1] - 1]}, {[1]}], 0 <= a[1] f[2](a[1], 1, a[3], 1) = [{[a[1] + a[3] - 1]}, {[{[a[1]]}, {[2]}]}], 0 <= a[1], 1 <= a[3] f[2](a[1], a[2], a[3], 1) = [{[a[1] + a[3] - 1]}, {[1 + a[1]]}], 0 <= a[1], 2 <= a[2], 0 <= a[3] f[2](0, 0, a[3], a[4]) = [a[3] - 1], 2 <= a[4], 0 <= a[3] f[2](a[1], 0, a[3], a[4]) = [a[1] - 1/2 + a[3]], 1 <= a[1], 2 <= a[4], 0 <= a[3] f[2](a[1], a[2], a[3], a[4]) = [a[1] + a[3]], 0 <= a[1], 2 <= a[4], 1 <= a[2], 0 <= a[3] Theorem: ############################# Let , f[3], be the value of , [T $ a[1], F, T $ a[2], B, T $ a[3], B, T $ a[4], F] ############################# Then we have: f[3](0, 0, 0, 0) = [-2] f[3](1, 0, 0, 0) = [-1] f[3](2, 0, 0, 0) = [{[0]}, {[0]}] f[3](3, 0, 0, 0) = [{[1/2]}, {[0]}] f[3](a[1], 0, 0, 0) = [{[{[1]}, {[1]}]}, {[0]}], 4 <= a[1] f[3](a[1], a[2], 0, 0) = [-1 + a[2]], 0 <= a[1], 1 <= a[2] f[3](0, 0, 1, 0) = [-1/2] f[3](1, 0, 1, 0) = [{[0]}, {[0]}] f[3](a[1], 0, 1, 0) = [{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[0]}], 2 <= a[1] f[3](a[1], a[2], 1, 0) = [{[a[2]]}, {[a[2]]}], 0 <= a[1], 1 <= a[2] f[3](a[1], a[2], a[3], 0) = [{[{[a[1] + 2 a[2] + a[3] - 2]}, {[1 + a[2]]}]}, {[a[2]]}], 0 <= a[1], 0 <= a[2], 2 <= a[3] f[3](0, 0, 0, 1) = [-1] f[3](1, 0, 0, 1) = [0] f[3](a[1], 0, 0, 1) = [{[{[a[1] - 2]}, {[2]}]}, {[1]}], 2 <= a[1] f[3](a[1], a[2], 0, 1) = [a[2]], 0 <= a[1], 1 <= a[2] f[3](a[1], a[2], a[3], 1) = [{[a[1] + 2 a[2] + a[3] - 1]}, {[1 + a[2]]}], 0 <= a[1], 0 <= a[2], 1 <= a[3] f[3](a[1], a[2], a[3], a[4]) = [a[1] + 2 a[2] + a[3]], 0 <= a[1], 0 <= a[2], 2 <= a[4], 0 <= a[3] Theorem: ############################# Let , f[4], be the value of , [T $ a[1], B, T $ a[2], F, T $ a[3], F, B] ############################# Then we have: f[4](0, 0, 0) = [-2] f[4](1, 0, 0) = [{[0]}, {[-1/2]}] f[4](a[1], 0, 0) = [{[{[a[1] - 2]}, {[1/2]}]}, {[{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}]}], 2 <= a[1] f[4](0, 0, a[3]) = [-1], 1 <= a[3] f[4](a[1], 0, a[3]) = [{[{[a[1] - 2]}, {[1]}]}, {[0]}], 1 <= a[1], 1 <= a[3] f[4](a[1], 1, 0) = [{[a[1] - 1]}, {[1/2]}], 0 <= a[1] f[4](a[1], 1, a[3]) = [{[a[1] - 1]}, {[1]}], 0 <= a[1], 1 <= a[3] f[4](a[1], a[2], a[3]) = [a[1]], 0 <= a[1], 2 <= a[2], 0 <= a[3] Theorem: ############################# Let , f[5], be the value of , [T $ a[1], F, T $ a[2], B, T $ a[3], F, B] ############################# Then we have: f[5](0, 0, 0) = [-1] f[5](1, 0, 0) = [-1/2] f[5](a[1], 0, 0) = [{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}], 2 <= a[1] f[5](a[1], a[2], 0) = [- 1/2 + a[2]], 0 <= a[1], 1 <= a[2] f[5](a[1], a[2], a[3]) = [a[2]], 0 <= a[1], 0 <= a[2], 1 <= a[3] Theorem: ############################# Let , f[6], be the value of , [T $ a[1], F, T $ a[2], F, B, B] ############################# Then we have: f[6](a[1], a[2]) = [a[2]], 0 <= a[1], 0 <= a[2] Now we will prove by applying induction to each of the above conjectures one by one. If everything is true, we get the complete proofs. ############### ##Begin to prove## ############### ########## #, f[1], # ########## For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 0, 0)}] = , [{}, {[-3]}] = , [-4] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, 0)}, {f[2](1, 0, 0, 0)}] = , [{[-3/2]}, {[{[-1]}, {[-1]}]}] = , [-1] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](1, 1, 0, 0)}, {f[2](2, 0, 0, 0)}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](2, 1, 0, 0)}, {f[2](3, 0, 0, 0)}] = , [{[{[1]}, {[{[1/2]}, {[0]}]}]}, {[0]}] = , [{[1/2]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 0, 0)}, {f[2](a[1], 0, 0, 0)}] = , [{[a[1] - 5/2]}, {[0]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 1, 0)}, {f[2](0, 1, 0, 0)}] = , [{[-2]}, {[-1]}] = , [-3/2] For the domain, {a[3] = 0, a[4] = 0, a[1] = 1, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 2, 0, 0), f[1](1, 0, 1, 0)}, {f[2](1, 1, 0, 0)}] = , [ {[{[0]}, {[0]}], [{[{[0]}, {[{[0]}, {[0]}]}]}, {[-1/4]}]}, {[{[0]}, {[0]}]} ] = , [0] For the domain, {a[3] = 0, a[4] = 0, a[1] = 2, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](1, 2, 0, 0), f[1](2, 0, 1, 0)}, {f[2](2, 1, 0, 0)}] = , [{[1], [{[1/2]}, {[1/2]}]}, {[{[1/2]}, {[0]}]}] = , [{[1]}, {[{[1/2]}, {[0]}]}] For the domain, {a[3] = 0, a[4] = 0, a[2] = 1, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](3, 0, 1, 0), f[1](2, 2, 0, 0)}, {f[2](3, 1, 0, 0)}] = , [{[1]}, {[{[3/2]}, {[0]}]}] = , [3/2] For the domain, {a[3] = 0, a[4] = 0, a[2] = 1, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1], 0, 1, 0), f[1](a[1] - 1, 2, 0, 0)}, {f[2](a[1], 1, 0, 0)}] = , [{[a[1] - 2], [{[{[a[1] - 2]}, {[a[1] - 2]}]}, {[{[1]}, {[1]}]}]}, {[{[a[1] - 3/2]}, {[0]}]}] = , [a[1] - 3/2] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 1, 0)}, {f[2](0, 2, 0, 0)}] = , [{[{[0]}, {[{[0]}, {[0]}]}]}, {[-1/4]}] For the domain, {a[3] = 0, a[4] = 0, a[1] = 1, a[2] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 3, 0, 0), f[1](1, 1, 1, 0)}, {f[2](1, 2, 0, 0)}] = , [{[{[0]}, {[0]}], [{[2]}, {[{[1/2]}, {[1/2]}]}]}, {[1/2]}] = , [{[1/2]}, {[1/2]}] For the domain, {a[3] = 0, a[4] = 0, a[2] = 2, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 3, 0, 0), f[1](a[1], 1, 1, 0)}, {f[2](a[1], 2, 0, 0)}] = , [{[{[a[1] - 1]}, {[a[1] - 1]}], [{[2 a[1]]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}]}, {[a[1] - 1/2]}] = , [a[1] - 1] For the domain, {a[3] = 0, a[4] = 0, 1 <= a[1], a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 4, 0, 0), f[1](a[1], 2, 1, 0)}, {f[2](a[1], 3, 0, 0)}] = , [{[{[a[1] - 1]}, {[a[1] - 1]}], [{[2 a[1] + 1]}, {[{[a[1]]}, {[a[1]]}]}]}, { [{[{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}]}, {[a[1]]}] }] = , [{[a[1]]}, {[a[1]]}] For the domain, {a[3] = 0, a[4] = 0, 1 <= a[1], 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 0, 0), f[1](a[1], -1 + a[2], 1, 0)}, {f[2](a[1], a[2], 0, 0)}] = , [{[{[a[1] - 1]}, {[a[1] - 1]}], [{[2 a[1] + a[2] - 2]}, {[{[a[1]]}, {[a[1]]}]}]}, {[{[{[{[2 a[1] + a[2] - 3]}, {[a[1] + 1/2]}]}, {[a[1]]}]}, {[a[1]]}]}] = , [{[a[1]]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 2, 1, 0)}, {f[2](0, 3, 0, 0)}] = , [{[{[1]}, {[{[0]}, {[0]}]}]}, {[{[{[{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}]}, {[0]}]}, {[0]}]}] = , [{[0]}, {[0]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 1, 0)}, {f[2](0, a[2], 0, 0)}] = , [{[{[-2 + a[2]]}, {[{[0]}, {[0]}]}]}, {[{[{[{[-3 + a[2]]}, {[1/2]}]}, {[0]}]}, {[0]}]}] = , [{[0]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[4] = 0, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 1, 0)}] = , [{}, {[-3/2]}] = , [-2] For the domain, {a[2] = 0, a[4] = 0, a[1] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 1, 0)}, {f[2](1, 0, 1, 0)}] = , [{[{[0]}, {[{[0]}, {[0]}]}]}, {[0]}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, a[4] = 0, a[1] = 2, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](1, 1, 1, 0)}, {f[2](2, 0, 1, 0)}] = , [{[{[2]}, {[{[1/2]}, {[1/2]}]}]}, {[{[1]}, {[{[1/2]}, {[0]}]}]}] = , [1] For the domain, {a[2] = 0, a[4] = 0, a[3] = 1, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](2, 1, 1, 0)}, {f[2](3, 0, 1, 0)}] = , [{[{[4]}, {[{[{[1]}, {[1]}]}, {[{[1]}, {[1]}]}]}]}, {[{[1]}, {[1]}]}] = , [1] For the domain, {a[2] = 0, a[4] = 0, a[3] = 1, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 1, 0)}, {f[2](a[1], 0, 1, 0)}] = , [{[{[2 a[1] - 2]}, {[{[{[a[1] - 2]}, {[a[1] - 2]}]}, {[{[1]}, {[1]}]}]}]}, {[{[1]}, {[1]}]}] = , [{[{[a[1] - 2]}, {[a[1] - 2]}]}, {[{[1]}, {[1]}]}] For the domain, {a[1] = 0, a[4] = 0, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 2, 0)}, {f[2](0, 1, 1, 0)}] = , [{[0]}, {[{[0]}, {[0]}]}] For the domain, {a[1] = 0, a[4] = 0, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 2, 0)}, {f[2](0, 1, 1, 0)}] = , [{[0]}, {[{[0]}, {[0]}]}] For the domain, {a[4] = 0, a[1] = 1, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 2, 1, 0), f[1](1, 0, 2, 0)}, {f[2](1, 1, 1, 0)}] = , [{[2], [{[1]}, {[{[0]}, {[0]}]}]}, {[{[1/2]}, {[1/2]}]}] = , [{[2]}, {[{[1/2]}, {[1/2]}]}] For the domain, {a[4] = 0, a[1] = 1, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 2, 1, 0), f[1](1, 0, 2, 0)}, {f[2](1, 1, 1, 0)}] = , [{[2], [{[1]}, {[{[0]}, {[0]}]}]}, {[{[1/2]}, {[1/2]}]}] = , [{[2]}, {[{[1/2]}, {[1/2]}]}] For the domain, {a[4] = 0, a[2] = 1, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 2, 1, 0), f[1](a[1], 0, 2, 0)}, {f[2](a[1], 1, 1, 0)}] = , [{[2 a[1]], [{[2 a[1] - 1]}, {[{[a[1] - 1]}, {[a[1] - 1]}]}]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}] = , [{[2 a[1]]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}] For the domain, {a[4] = 0, a[2] = 1, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 2, 1, 0), f[1](a[1], 0, 2, 0)}, {f[2](a[1], 1, 1, 0)}] = , [{[2 a[1]], [{[2 a[1] - 1]}, {[{[a[1] - 1]}, {[a[1] - 1]}]}]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}] = , [{[2 a[1]]}, {[{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}] For the domain, {a[4] = 0, a[3] = 1, 2 <= a[2], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, 0), f[1](a[1], -1 + a[2], 2, 0)}, {f[2](a[1], a[2], 1, 0)}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[{[a[1] - 1]}, {[a[1] - 1]}]}]}, {[{[{[a[1] - 1]}, {[1 + a[1]]}]}, {[a[1]]}]}] = , [{[2 a[1] + a[2] - 1]}, {[{[a[1]]}, {[a[1]]}]}] For the domain, {a[4] = 0, a[3] = 1, 2 <= a[2], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, 0), f[1](a[1], -1 + a[2], 2, 0)}, {f[2](a[1], a[2], 1, 0)}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[{[a[1] - 1]}, {[a[1] - 1]}]}]}, {[{[{[a[1] - 1]}, {[1 + a[1]]}]}, {[a[1]]}]}] = , [{[2 a[1] + a[2] - 1]}, {[{[a[1]]}, {[a[1]]}]}] For the domain, {a[1] = 0, a[4] = 0, a[3] = 1, 2 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, 0)}, {f[2](0, a[2], 1, 0)}] = , [{[-1 + a[2]]}, {[{[{[-1]}, {[1]}]}, {[0]}]}] = , [{[-1 + a[2]]}, {[{[0]}, {[0]}]}] For the domain, {a[1] = 0, a[4] = 0, a[3] = 1, 2 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, 0)}, {f[2](0, a[2], 1, 0)}] = , [{[-1 + a[2]]}, {[{[{[-1]}, {[1]}]}, {[0]}]}] = , [{[-1 + a[2]]}, {[{[0]}, {[0]}]}] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {a[2] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, a[3], a[4])}, {}] = , [{[2 a[1] - 1]}, {}] = , [2 a[1]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {a[2] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, a[3], a[4])}, {}] = , [{[2 a[1] - 1]}, {}] = , [2 a[1]] For the domain, {a[1] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], a[3] + 1, a[4])}, {}] = , [{[-1 + a[2]]}, {}] = , [a[2]] For the domain, {a[1] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], a[3] + 1, a[4])}, {}] = , [{[-1 + a[2]]}, {}] = , [a[2]] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[3], 0 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {}] = , [{}, {}] = , [0] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 0, 1)}] = , [{}, {[-2]}] = , [-3] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, 1)}, {f[2](1, 0, 0, 1)}] = , [{[-1/2]}, {[{[0]}, {[0]}]}] = , [0] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 0, 1)}, {f[2](a[1], 0, 0, 1)}] = , [{[{[a[1] - 3/2]}, {[{[a[1] - 2]}, {[1]}]}]}, {[1]}] = , [{[a[1] - 2]}, {[1]}] For the domain, {a[1] = 0, a[3] = 0, a[2] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 1, 1)}, {f[2](0, 1, 0, 1)}] = , [{[-1]}, {[{[-1]}, {[1]}]}] = , [-1/2] For the domain, {a[3] = 0, a[2] = 1, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 2, 0, 1), f[1](a[1], 0, 1, 1)}, {f[2](a[1], 1, 0, 1)}] = , [{[ {[{[2 a[1] - 2]}, {[{[a[1] - 1]}, {[{[a[1] - 1]}, {[2]}]}]}]}, {[a[1] - 1]} ], [a[1] - 1/2]}, {[{[a[1] - 1]}, {[1]}]}] = , [{[a[1] - 1/2]}, {[{[a[1] - 1]}, {[1]}]}] For the domain, {a[3] = 0, a[2] = 2, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1], 1, 1, 1), f[1](a[1] - 1, 3, 0, 1)}, {f[2](a[1], 2, 0, 1)}] = , [{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}], [{[{[2 a[1] - 1]}, {[a[1] - 1/2]}]}, {[a[1] - 1]}]}, {[{[a[1] - 1]}, {[1 + a[1]]}]}] = , [{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, a[2] = 2, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 1, 1)}, {f[2](0, 2, 0, 1)}] = , [{[{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}]}, {[{[-1]}, {[1]}]}] = , [{[1/4]}, {[0]}] For the domain, {a[3] = 0, 3 <= a[2], a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 0, 1), f[1](a[1], -1 + a[2], 1, 1)}, {f[2](a[1], a[2], 0, 1)}] = , [{[{[2 a[1] + a[2] - 2]}, {[a[1] + 1/2]}], [{[{[2 a[1] + a[2] - 3]}, {[a[1] - 1/2]}]}, {[a[1] - 1]}]}, {[{[a[1] - 1]}, {[1 + a[1]]}]}] = , [{[{[2 a[1] + a[2] - 2]}, {[a[1] + 1/2]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, 3 <= a[2], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 1, 1)}, {f[2](0, a[2], 0, 1)}] = , [{[{[-2 + a[2]]}, {[1/2]}]}, {[{[-1]}, {[1]}]}] = , [{[{[-2 + a[2]]}, {[1/2]}]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 1, 1)}] = , [{}, {[{[-1]}, {[0]}]}] = , [-1] For the domain, {a[2] = 0, a[3] = 1, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 1, 1)}, {f[2](a[1], 0, 1, 1)}] = , [{[{[2 a[1] - 2]}, {[{[a[1] - 1]}, {[{[a[1] - 1]}, {[2]}]}]}]}, {[{[a[1] - 1/2]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}]}] = , [a[1] - 1/2] For the domain, {a[2] = 1, a[3] = 1, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 2, 1, 1), f[1](a[1], 0, 2, 1)}, {f[2](a[1], 1, 1, 1)}] = , [{[2 a[1]], [{[2 a[1] - 1]}, {[a[1] - 1/2]}]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}] = , [{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}] For the domain, {a[2] = 1, a[3] = 1, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 2, 1, 1), f[1](a[1], 0, 2, 1)}, {f[2](a[1], 1, 1, 1)}] = , [{[2 a[1]], [{[2 a[1] - 1]}, {[a[1] - 1/2]}]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}] = , [{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}] For the domain, {a[1] = 0, a[2] = 1, a[3] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 2, 1)}, {f[2](0, 1, 1, 1)}] = , [{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}] = , [1/4] For the domain, {a[1] = 0, a[2] = 1, a[3] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 2, 1)}, {f[2](0, 1, 1, 1)}] = , [{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}] = , [1/4] For the domain, {a[3] = 1, 2 <= a[2], a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, 1), f[1](a[1], -1 + a[2], 2, 1)}, {f[2](a[1], a[2], 1, 1)}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[a[1] - 1/2]}]}, {[{[a[1]]}, {[1 + a[1]]}]}] = , [{[2 a[1] + a[2] - 1]}, {[a[1] + 1/2]}] For the domain, {a[3] = 1, 2 <= a[2], a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, 1), f[1](a[1], -1 + a[2], 2, 1)}, {f[2](a[1], a[2], 1, 1)}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[a[1] - 1/2]}]}, {[{[a[1]]}, {[1 + a[1]]}]}] = , [{[2 a[1] + a[2] - 1]}, {[a[1] + 1/2]}] For the domain, {a[1] = 0, a[3] = 1, 2 <= a[2], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, 1)}, {f[2](0, a[2], 1, 1)}] = , [{[-1 + a[2]]}, {[{[0]}, {[1]}]}] = , [{[-1 + a[2]]}, {[1/2]}] For the domain, {a[1] = 0, a[3] = 1, 2 <= a[2], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, 1)}, {f[2](0, a[2], 1, 1)}] = , [{[-1 + a[2]]}, {[{[0]}, {[1]}]}] = , [{[-1 + a[2]]}, {[1/2]}] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {a[2] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, a[3], a[4])}, {}] = , [{[2 a[1] - 1]}, {}] = , [2 a[1]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {2 <= a[3], 0 <= a[4], 1 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[1](a[1] - 1, 1 + a[2], a[3], a[4]), f[1](a[1], -1 + a[2], a[3] + 1, a[4]) }, {}] = , [{[2 a[1] + a[2] - 1]}, {}] = , [2 a[1] + a[2]] For the domain, {a[2] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, a[3], a[4])}, {}] = , [{[2 a[1] - 1]}, {}] = , [2 a[1]] For the domain, {a[1] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], a[3] + 1, a[4])}, {}] = , [{[-1 + a[2]]}, {}] = , [a[2]] For the domain, {a[1] = 0, 2 <= a[3], 0 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], a[3] + 1, a[4])}, {}] = , [{[-1 + a[2]]}, {}] = , [a[2]] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[3], 0 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {}] = , [{}, {}] = , [0] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 0, a[4])}] = , [{}, {[-1]}] = , [-2] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 0, a[4])}] = , [{}, {[-1]}] = , [-2] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, a[4])}, {f[2](1, 0, 0, a[4])}] = , [{[-1/2]}, {[1/2]}] = , [0] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 0, a[4])}, {f[2](a[1], 0, 0, a[4])}] = , [{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[a[1] - 1/2]}] = , [a[1] - 1] For the domain, {a[1] = 0, a[3] = 0, a[2] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 1, a[4])}, {f[2](0, 1, 0, a[4])}] = , [{[-1]}, {[0]}] = , [-1/2] For the domain, {a[3] = 0, a[2] = 1, 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 2, 0, a[4]), f[1](a[1], 0, 1, a[4])}, {f[2](a[1], 1, 0, a[4])}] = , [{[{[{[2 a[1] - 2]}, {[a[1]]}]}, {[a[1] - 1]}], [a[1]]}, {[a[1]]}] = , [{[a[1]]}, {[a[1]]}] For the domain, {a[3] = 0, 2 <= a[2], 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1], -1 + a[2], 1, a[4]), f[1](a[1] - 1, 1 + a[2], 0, a[4])}, {f[2](a[1], a[2], 0, a[4])}] = , [{[{[2 a[1] + a[2] - 2]}, {[1 + a[1]]}], [{[{[2 a[1] + a[2] - 3]}, {[a[1]]}]}, {[a[1] - 1]}]}, {[a[1]]}] = , [{[{[2 a[1] + a[2] - 2]}, {[1 + a[1]]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, 2 <= a[2], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 1, a[4])}, {f[2](0, a[2], 0, a[4])}] = , [{[{[-2 + a[2]]}, {[1]}]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 1, a[4])}] = , [{}, {[0]}] = , [-1] For the domain, {a[1] = 0, a[2] = 0, a[3] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[2](0, 0, 1, a[4])}] = , [{}, {[0]}] = , [-1] For the domain, {a[2] = 0, a[3] = 1, 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1] - 1, 1, 1, a[4])}, {f[2](a[1], 0, 1, a[4])}] = , [{[{[2 a[1] - 2]}, {[a[1]]}]}, {[a[1] + 1/2]}] = , [a[1]] For the domain, {a[3] = 1, 1 <= a[1], 2 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, a[4]), f[1](a[1], -1 + a[2], 2, a[4])}, {f[2](a[1], a[2], 1, a[4])}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[a[1]]}]}, {[1 + a[1]]}] = , [{[2 a[1] + a[2] - 1]}, {[1 + a[1]]}] For the domain, {a[3] = 1, 1 <= a[1], 2 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[1](a[1] - 1, 1 + a[2], 1, a[4]), f[1](a[1], -1 + a[2], 2, a[4])}, {f[2](a[1], a[2], 1, a[4])}] = , [{[2 a[1] + a[2] - 1], [{[2 a[1] + a[2] - 2]}, {[a[1]]}]}, {[1 + a[1]]}] = , [{[2 a[1] + a[2] - 1]}, {[1 + a[1]]}] For the domain, {a[1] = 0, a[3] = 1, 2 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, a[4])}, {f[2](0, a[2], 1, a[4])}] = , [{[-1 + a[2]]}, {[1]}] For the domain, {a[1] = 0, a[3] = 1, 2 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 2, a[4])}, {f[2](0, a[2], 1, a[4])}] = , [{[-1 + a[2]]}, {[1]}] ########## #, f[2], # ########## For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[3](0, 0, 0, 0), f[4](0, 0, 0) + [0]}] = , [{}, {[-2]}] = , [-3] For the domain, {a[1] = 0, a[2] = 0, a[4] = 0, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 0, 1)}, {f[3](0, 0, 1, 0), f[4](0, 0, 1) + [0]}] = , [{[-2]}, {[-1], [-1/2]}] = , [-3/2] For the domain, {a[1] = 0, a[2] = 0, a[4] = 0, 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, 1)}, {f[4](0, 0, a[3]) + [0], f[3](0, 0, a[3], 0)}] = , [{[{[a[3] - 3]}, {[0]}]}, {[-1], [{[{[a[3] - 2]}, {[1]}]}, {[0]}]}] = , [{[{[a[3] - 3]}, {[0]}]}, {[-1]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[4](0, 0, 0) + [1], f[3](0, 0, 0, 1)}] = , [{}, {[-1]}] = , [-2] For the domain, {a[1] = 0, a[2] = 0, a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, 2)}, {f[4](0, 0, a[3]) + [1], f[3](0, 0, a[3], 1)}] = , [{[a[3] - 2]}, {[0], [{[a[3] - 1]}, {[1]}]}] = , [{[a[3] - 2]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, 2)}, {f[4](0, 0, a[3]) + [1], f[3](0, 0, a[3], 1)}] = , [{[a[3] - 2]}, {[0], [{[a[3] - 1]}, {[1]}]}] = , [{[a[3] - 2]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[3](0, 0, 0, a[4])}] = , [{}, {[0]}] = , [-1] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, a[4] + 1)}, {f[3](0, 0, a[3], a[4])}] = , [{[a[3] - 2]}, {[a[3]]}] = , [a[3] - 1] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, a[4] + 1)}, {f[3](0, 0, a[3], a[4])}] = , [{[a[3] - 2]}, {[a[3]]}] = , [a[3] - 1] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 0, 0)}, {f[4](1, 0, 0) + [0], f[3](1, 0, 0, 0)}] = , [{[-1]}, {[-1], [{[0]}, {[-1/2]}]}] = , [{[-1]}, {[-1]}] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 1, 0, 0)}, {f[4](2, 0, 0) + [0], f[3](2, 0, 0, 0)}] = , [{[{[0]}, {[0]}]}, {[{[0]}, {[0]}], [{[{[0]}, {[1/2]}]}, {[{[{[{[-1]}, {[1]}]}, {[0]}]}, {[0]}]}]}] = , [0] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](2, 1, 0, 0)}, {f[3](3, 0, 0, 0), f[4](3, 0, 0) + [0]}] = , [{[{[1/2]}, {[0]}]}, {[{[1/2]}, {[0]}], [{[{[1]}, {[1/2]}]}, {[{[{[{[0]}, {[1]}]}, {[0]}]}, {[0]}]}]}] = , [0] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 1, 0, 0)}, {f[4](a[1], 0, 0) + [0], f[3](a[1], 0, 0, 0)}] = , [{[{[a[1] - 5/2]}, {[0]}]}, {[{[{[1]}, {[1]}]}, {[0]}], [{[{[a[1] - 2]}, {[1/2]}]}, {[{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}]}]} ] = , [0] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 0, 1)}, {f[4](0, 1, 0) + [0], f[3](0, 1, 0, 0)}] = , [{[-3]}, {[0], [{[-1]}, {[1/2]}]}] = , [-1] For the domain, {a[3] = 0, a[4] = 0, a[1] = 1, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 2, 0, 0), f[1](1, 0, 0, 1)}, {f[4](1, 1, 0) + [0], f[3](1, 1, 0, 0)}] = , [{[-1/4], [{[-1]}, {[1]}]}, {[0], [{[0]}, {[1/2]}]}] = , [{[0]}, {[0]}] For the domain, {a[3] = 0, a[4] = 0, a[2] = 1, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 2, 0, 0), f[1](a[1], 0, 0, 1)}, {f[4](a[1], 1, 0) + [0], f[3](a[1], 1, 0, 0)}] = , [{[a[1] - 3/2], [{[a[1] - 2]}, {[1]}]}, {[0], [{[a[1] - 1]}, {[1/2]}]}] = , [{[a[1] - 3/2]}, {[0]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, 1)}, {f[4](0, 2, 0) + [0]}] = , [{[-1/2]}, {[0]}] = , [-1/4] For the domain, {a[3] = 0, a[4] = 0, a[2] = 2, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 3, 0, 0), f[1](a[1], 1, 0, 1)}, {f[4](a[1], 2, 0) + [0]}] = , [{[{[ {[{[2 a[1] - 2]}, {[{[a[1] - 1]}, {[{[a[1] - 1]}, {[2]}]}]}]}, {[a[1] - 1]} ]}, {[a[1] - 1]}], [{[a[1] - 1/2]}, {[{[a[1] - 1]}, {[1]}]}]}, {[a[1]]}] = , [a[1] - 1/2] For the domain, {a[3] = 0, a[4] = 0, 1 <= a[1], a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 4, 0, 0), f[1](a[1], 2, 0, 1)}, {f[4](a[1], 3, 0) + [0]}] = , [{[{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}], [{[{[{[2 a[1] - 1]}, {[a[1] - 1/2]}]}, {[a[1] - 1]}]}, {[a[1] - 1]}]}, {[a[1]]}] = , [{[{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 2, 0, 1)}, {f[4](0, 3, 0) + [0]}] = , [{[{[{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}]}, {[0]}]}, {[0]}] = , [{[{[1/4]}, {[0]}]}, {[0]}] For the domain, {a[3] = 0, a[4] = 0, 1 <= a[1], 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1 + a[2], 0, 0), f[1](a[1], -1 + a[2], 0, 1)}, {f[4](a[1], a[2], 0) + [0]}] = , [{[{[{[2 a[1] + a[2] - 3]}, {[a[1] + 1/2]}]}, {[a[1]]}], [{[{[{[2 a[1] - 4 + a[2]]}, {[a[1] - 1/2]}]}, {[a[1] - 1]}]}, {[a[1] - 1]}] }, {[a[1]]}] = , [{[{[{[2 a[1] + a[2] - 3]}, {[a[1] + 1/2]}]}, {[a[1]]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 0, 1)}, {f[4](0, a[2], 0) + [0]}] = , [{[{[{[-3 + a[2]]}, {[1/2]}]}, {[0]}]}, {[0]}] For the domain, {a[2] = 0, a[4] = 0, a[1] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 1, 0), f[2](1, 0, 0, 1)}, {f[3](1, 0, 1, 0), f[4](1, 0, 1) + [0]}] = , [{[{[0]}, {[0]}]}, {[{[0]}, {[0]}], [{[{[-1]}, {[1]}]}, {[0]}]}] = , [0] For the domain, {a[2] = 0, a[4] = 0, a[1] = 2, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 1, 1, 0), f[2](2, 0, 0, 1)}, {f[3](2, 0, 1, 0), f[4](2, 0, 1) + [0]}] = , [{[1], [{[1/2]}, {[1/2]}]}, {[{[{[0]}, {[1]}]}, {[0]}], [{[{[{[0]}, {[2]}]}, {[1]}]}, {[0]}]}] = , [{[1]}, {[{[1/2]}, {[0]}]}] For the domain, {a[2] = 0, a[4] = 0, 3 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 1, 1, 0), f[2](a[1], 0, 0, 1)}, {f[4](a[1], 0, 1) + [0], f[3](a[1], 0, 1, 0)}] = , [{[1], [{[{[a[1] - 2]}, {[a[1] - 2]}]}, {[{[1]}, {[1]}]}]}, { [{[{[a[1] - 2]}, {[1]}]}, {[0]}], [{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[0]}]}] = , [{[1]}, {[1]}] For the domain, {a[2] = 0, a[4] = 0, 2 <= a[3], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1, a[3], 0), f[2](a[1], 0, a[3] - 1, 1)}, {f[4](a[1], 0, a[3]) + [0], f[3](a[1], 0, a[3], 0)}] = , [{[{[a[1] + a[3] - 5/2]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}], [ {[{[a[1] - 3 + a[3]]}, {[{[a[1] - 1]}, {[2]}]}]}, {[2], [{[a[1] - 2]}, {[1]}]}]}, {[{[{[a[1] - 2]}, {[1]}]}, {[0]}], [{[{[a[1] + a[3] - 2]}, {[1]}]}, {[0]}]} ] = , [{[{[a[1] + a[3] - 5/2]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}]}, {[{[{[a[1] - 2]}, {[1]}]}, {[0]}]}] For the domain, {a[1] = 0, a[4] = 0, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 0, 1)}, {f[3](0, 1, 1, 0), f[4](0, 1, 1) + [0]}] = , [{[{[-1]}, {[1]}]}, {[{[1]}, {[1]}], [{[-1]}, {[1]}]}] = , [{[0]}, {[0]}] For the domain, {a[4] = 0, a[1] = 1, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 2, 1, 0), f[2](1, 1, 0, 1)}, {f[4](1, 1, 1) + [0], f[3](1, 1, 1, 0)}] = , [{[{[0]}, {[1]}], [{[{[-1]}, {[1]}]}, {[0]}]}, {[{[1]}, {[1]}], [{[0]}, {[1]}]}] = , [{[1/2]}, {[1/2]}] For the domain, {a[4] = 0, a[2] = 1, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 2, 1, 0), f[2](a[1], 1, 0, 1)}, {f[4](a[1], 1, 1) + [0], f[3](a[1], 1, 1, 0)}] = , [{[{[a[1] - 1]}, {[1]}], [{[{[a[1] - 2]}, {[a[1]]}]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}], [{[a[1] - 1]}, {[1]}]}] = , [{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}] For the domain, {a[4] = 0, a[2] = 1, 2 <= a[3], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 2, a[3], 0), f[2](a[1], 1, a[3] - 1, 1)}, {f[4](a[1], 1, a[3]) + [0], f[3](a[1], 1, a[3], 0)}] = , [{[{[a[1] + a[3] - 2]}, {[{[a[1]]}, {[2]}]}], [{[{[a[1] - 3 + a[3]]}, {[a[1]]}]}, {[a[1] - 1]}]}, {[{[a[1] - 1]}, {[1]}], [{[{[a[1] + a[3]]}, {[2]}]}, {[1]}]}] = , [ {[{[a[1] + a[3] - 2]}, {[{[a[1]]}, {[2]}]}]}, {[2], [{[a[1] - 1]}, {[1]}]}] For the domain, {a[1] = 0, a[4] = 0, a[2] = 1, 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, a[3] - 1, 1)}, {f[4](0, 1, a[3]) + [0], f[3](0, 1, a[3], 0)}] = , [{[{[a[3] - 2]}, {[{[0]}, {[2]}]}]}, {[{[-1]}, {[1]}], [{[{[a[3]]}, {[2]}]}, {[1]}]}] = , [{[{[a[3] - 2]}, {[1]}]}, {[0]}] For the domain, {a[4] = 0, 2 <= a[2], 1 <= a[1], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1 + a[2], a[3], 0), f[2](a[1], a[2], a[3] - 1, 1)}, {f[4](a[1], a[2], a[3]) + [0]}] = , [{[{[{[a[1] - 3 + a[3]]}, {[a[1]]}]}, {[a[1] - 1]}], [{[a[1] + a[3] - 2]}, {[1 + a[1]]}]}, {[a[1]]}] = , [{[{[a[1] + a[3] - 2]}, {[1 + a[1]]}]}, {[a[1]]}] For the domain, {a[1] = 0, a[4] = 0, 2 <= a[2], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, a[2], a[3] - 1, 1)}, {f[4](0, a[2], a[3]) + [0]}] = , [{[{[a[3] - 2]}, {[1]}]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 0, 1)}, {f[4](1, 0, 0) + [1], f[3](1, 0, 0, 1)}] = , [{[{[-1]}, {[1]}]}, {[0], [{[1]}, {[1/2]}]}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 1, 0, 1)}, {f[4](a[1], 0, 0) + [1], f[3](a[1], 0, 0, 1)}] = , [{[{[a[1] - 2]}, {[1]}]}, {[{[{[a[1] - 2]}, {[2]}]}, {[1]}], [{[{[a[1] - 1]}, {[3/2]}]}, {[{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[1]}]}]} ] = , [1] For the domain, {a[2] = 0, a[4] = 1, 1 <= a[1], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1, a[3], 1), f[2](a[1], 0, a[3] - 1, 2)}, {f[4](a[1], 0, a[3]) + [1], f[3](a[1], 0, a[3], 1)}] = , [{[a[1] - 3/2 + a[3]], [{[a[1] + a[3] - 2]}, {[{[a[1] - 1]}, {[2]}]}]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}], [{[a[1] + a[3] - 1]}, {[1]}]}] = , [{[a[1] - 3/2 + a[3]]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}] For the domain, {a[3] = 0, a[2] = 1, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1], 0, 0, 2), f[2](a[1] - 1, 2, 0, 1)}, {f[3](a[1], 1, 0, 1), f[4](a[1], 1, 0) + [1]}] = , [{[{[a[1] - 2]}, {[a[1]]}], [a[1] - 1]}, {[{[a[1]]}, {[3/2]}], [1]}] = , [{[a[1] - 1]}, {[1]}] For the domain, {a[1] = 0, a[3] = 0, a[2] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 0, 2)}, {f[3](0, 1, 0, 1), f[4](0, 1, 0) + [1]}] = , [{[-2]}, {[1], [{[0]}, {[3/2]}]}] = , [0] For the domain, {a[2] = 1, a[4] = 1, 1 <= a[1], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1], 1, a[3] - 1, 2), f[2](a[1] - 1, 2, a[3], 1)}, {f[3](a[1], 1, a[3], 1), f[4](a[1], 1, a[3]) + [1]}] = , [{[{[a[1] + a[3] - 2]}, {[a[1]]}], [a[1] + a[3] - 1]}, {[{[a[1]]}, {[2]}], [{[a[1] + 1 + a[3]]}, {[2]}]}] = , [{[a[1] + a[3] - 1]}, {[{[a[1]]}, {[2]}]}] For the domain, {a[1] = 0, a[2] = 1, a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, a[3] - 1, 2)}, {f[4](0, 1, a[3]) + [1], f[3](0, 1, a[3], 1)}] = , [{[a[3] - 1]}, {[{[0]}, {[2]}], [{[a[3] + 1]}, {[2]}]}] = , [{[a[3] - 1]}, {[1]}] For the domain, {a[3] = 0, a[2] = 2, a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 3, 0, 1), f[1](a[1], 1, 0, 2)}, {f[4](a[1], 2, 0) + [1]}] = , [{[{[a[1]]}, {[a[1]]}], [{[a[1] - 2]}, {[a[1]]}]}, {[1 + a[1]]}] = , [a[1]] For the domain, {a[3] = 0, 3 <= a[2], a[4] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1 + a[2], 0, 1), f[1](a[1], -1 + a[2], 0, 2)}, {f[4](a[1], a[2], 0) + [1]}] = , [{[{[{[2 a[1] + a[2] - 3]}, {[1 + a[1]]}]}, {[a[1]]}], [{[a[1] - 2]}, {[a[1]]}]}, {[1 + a[1]]}] = , [a[1]] For the domain, {a[1] = 0, a[3] = 0, a[2] = 2, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, 2)}, {f[4](0, 2, 0) + [1]}] = , [{[-1/2]}, {[1]}] = , [0] For the domain, {a[1] = 0, a[3] = 0, 3 <= a[2], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 0, 2)}, {f[4](0, a[2], 0) + [1]}] = , [{[{[{[-3 + a[2]]}, {[1]}]}, {[0]}]}, {[1]}] = , [0] For the domain, {2 <= a[2], a[4] = 1, 1 <= a[1], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1 + a[2], a[3], 1), f[2](a[1], a[2], a[3] - 1, 2)}, {f[4](a[1], a[2], a[3]) + [1]}] = , [{[a[1] + a[3] - 1], [{[a[1] + a[3] - 2]}, {[a[1]]}]}, {[1 + a[1]]}] = , [{[a[1] + a[3] - 1]}, {[1 + a[1]]}] For the domain, {a[1] = 0, 2 <= a[2], a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, a[2], a[3] - 1, 2)}, {f[4](0, a[2], a[3]) + [1]}] = , [{[a[3] - 1]}, {[1]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[3](0, 0, 0, a[4])}] = , [{}, {[0]}] = , [-1] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, a[4] + 1)}, {f[3](0, 0, a[3], a[4])}] = , [{[a[3] - 2]}, {[a[3]]}] = , [a[3] - 1] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, a[4] + 1)}, {f[3](0, 0, a[3], a[4])}] = , [{[a[3] - 2]}, {[a[3]]}] = , [a[3] - 1] For the domain, {a[2] = 0, a[3] = 0, 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 1, 0, a[4])}, {f[3](a[1], 0, 0, a[4])}] = , [{[a[1] - 1]}, {[a[1]]}] = , [a[1] - 1/2] For the domain, {a[2] = 0, 1 <= a[1], 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1], 0, a[3] - 1, a[4] + 1), f[2](a[1] - 1, 1, a[3], a[4])}, {f[3](a[1], 0, a[3], a[4])}] = , [{[a[1] - 3/2 + a[3]], [a[1] + a[3] - 1]}, {[a[1] + a[3]]}] = , [a[1] - 1/2 + a[3]] For the domain, {a[3] = 0, a[2] = 1, 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 2, 0, a[4]), f[1](a[1], 0, 0, a[4] + 1)}, {f[3](a[1], 1, 0, a[4])}] = , [{[a[1] - 1]}, {[a[1] + 2]}] = , [a[1]] For the domain, {a[1] = 0, a[3] = 0, a[2] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 0, 0, a[4] + 1)}, {f[3](0, 1, 0, a[4])}] = , [{[-2]}, {[2]}] = , [0] For the domain, {a[2] = 1, 1 <= a[1], 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 2, a[3], a[4]), f[2](a[1], 1, a[3] - 1, a[4] + 1)}, {f[3](a[1], 1, a[3], a[4])}] = , [{[a[1] + a[3] - 1]}, {[a[1] + 2 + a[3]]}] = , [a[1] + a[3]] For the domain, {a[1] = 0, a[2] = 1, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, a[3] - 1, a[4] + 1)}, {f[3](0, 1, a[3], a[4])}] = , [{[a[3] - 1]}, {[2 + a[3]]}] = , [a[3]] For the domain, {a[3] = 0, a[2] = 2, 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](a[1], 1, 0, a[4] + 1), f[2](a[1] - 1, 3, 0, a[4])}, {}] = , [{[a[1] - 1], [{[a[1]]}, {[a[1]]}]}, {}] = , [a[1]] For the domain, {a[3] = 0, 3 <= a[2], 1 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 1 + a[2], 0, a[4]), f[1](a[1], -1 + a[2], 0, a[4] + 1)}, {} ] = , [{[a[1] - 1], [{[{[2 a[1] + a[2] - 3]}, {[1 + a[1]]}]}, {[a[1]]}]}, {}] = , [a[1]] For the domain, {a[1] = 0, a[3] = 0, a[2] = 2, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, 1, 0, a[4] + 1)}, {}] = , [{[-1/2]}, {}] = , [0] For the domain, {a[1] = 0, a[3] = 0, 3 <= a[2], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[1](0, -1 + a[2], 0, a[4] + 1)}, {}] = , [{[{[{[-3 + a[2]]}, {[1]}]}, {[0]}]}, {}] = , [0] For the domain, {2 <= a[2], 1 <= a[1], 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[2](a[1], a[2], a[3] - 1, a[4] + 1), f[2](a[1] - 1, 1 + a[2], a[3], a[4])} , {}] = , [{[a[1] + a[3] - 1]}, {}] = , [a[1] + a[3]] For the domain, {a[1] = 0, 2 <= a[2], 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, a[2], a[3] - 1, a[4] + 1)}, {}] = , [{[a[3] - 1]}, {}] = , [a[3]] ########## #, f[3], # ########## For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[5](0, 0, 0) + [0]}] = , [{}, {[-1]}] = , [-2] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 1, 0)}, {f[5](1, 0, 0) + [0]}] = , [{[-3/2]}, {[-1/2]}] = , [-1] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 0, 1, 0)}, {f[5](2, 0, 0) + [0]}] = , [{[0]}, {[{[{[{[-1]}, {[1]}]}, {[0]}]}, {[0]}]}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](2, 0, 1, 0)}, {f[5](3, 0, 0) + [0]}] = , [{[{[1]}, {[{[1/2]}, {[0]}]}]}, {[{[{[{[0]}, {[1]}]}, {[0]}]}, {[0]}]}] = , [{[1/2]}, {[0]}] For the domain, {a[2] = 0, a[3] = 0, a[4] = 0, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 0, 1, 0)}, {f[5](a[1], 0, 0) + [0]}] = , [{[{[1]}, {[1]}]}, {[{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}]}] = , [{[{[1]}, {[1]}]}, {[0]}] For the domain, {a[1] = 0, a[3] = 0, a[4] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, 1, 0)}, {f[5](0, 1, 0) + [0]}] = , [{[-1/2]}, {[1/2]}] = , [0] For the domain, {a[3] = 0, a[4] = 0, a[1] = 1, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](1, 0, 1, 0)}, {f[5](1, 1, 0) + [0]}] = , [{[{[0]}, {[0]}]}, {[1/2]}] = , [0] For the domain, {a[3] = 0, a[4] = 0, a[2] = 1, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], 0, 1, 0)}, {f[5](a[1], 1, 0) + [0]}] = , [{[{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[0]}]}, {[1/2]}] = , [0] For the domain, {a[3] = 0, a[4] = 0, 0 <= a[1], 2 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], -1 + a[2], 1, 0)}, {f[5](a[1], a[2], 0) + [0]}] = , [{[{[-1 + a[2]]}, {[-1 + a[2]]}]}, {[- 1/2 + a[2]]}] = , [-1 + a[2]] For the domain, {a[1] = 0, a[2] = 0, a[4] = 0, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, 0, 1)}, {f[5](0, 0, 1) + [0]}] = , [{[-1]}, {[0]}] = , [-1/2] For the domain, {a[2] = 0, a[4] = 0, a[1] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 2, 0), f[3](1, 0, 0, 1)}, {f[5](1, 0, 1) + [0]}] = , [{[0], [{[{[-1]}, {[0]}]}, {[-1]}]}, {[0]}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, a[4] = 0, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 0, 2, 0), f[3](a[1], 0, 0, 1)}, {f[5](a[1], 0, 1) + [0]}] = , [{[{[{[a[1] - 2]}, {[2]}]}, {[1]}], [ {[{[a[1] - 3/2]}, {[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}]}, {[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}]}, {[0]}] = , [{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[0]}] For the domain, {a[4] = 0, 0 <= a[1], a[3] = 1, 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[3](a[1], -1 + a[2], 2, 0), f[3](a[1], a[2], 0, 1)}, {f[5](a[1], a[2], 1) + [0]}] = , [{[a[2]], [{[{[a[1] - 2 + 2 a[2]]}, {[a[2]]}]}, {[-1 + a[2]]}]}, {[a[2]]}] = , [{[a[2]]}, {[a[2]]}] For the domain, {a[2] = 0, a[4] = 0, a[1] = 1, 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](1, 0, a[3] - 1, 1), f[2](0, 0, a[3] + 1, 0)}, {f[5](1, 0, a[3]) + [0]}] = , [{[{[{[a[3] - 2]}, {[0]}]}, {[-1]}], [{[a[3] - 1]}, {[1]}]}, {[0]}] = , [{[{[a[3] - 1]}, {[1]}]}, {[0]}] For the domain, {a[2] = 0, a[4] = 0, 2 <= a[1], 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[3](a[1], 0, a[3] - 1, 1), f[2](a[1] - 1, 0, a[3] + 1, 0)}, {f[5](a[1], 0, a[3]) + [0]}] = , [{[{[{[a[1] + a[3] - 5/2]}, {[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}]}, {[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}], [{[a[1] + a[3] - 2]}, {[1]}]}, {[0]}] = , [{[{[a[1] + a[3] - 2]}, {[1]}]}, {[0]}] For the domain, {a[1] = 0, a[2] = 0, a[4] = 0, 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, a[3] - 1, 1)}, {f[5](0, 0, a[3]) + [0]}] = , [{[{[a[3] - 2]}, {[1]}]}, {[0]}] For the domain, {a[4] = 0, 0 <= a[1], 2 <= a[3], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[3](a[1], -1 + a[2], a[3] + 1, 0), f[3](a[1], a[2], a[3] - 1, 1)}, {f[5](a[1], a[2], a[3]) + [0]}] = , [{[{[a[1] + 2 a[2] + a[3] - 2]}, {[1 + a[2]]}], [{[{[a[1] - 3 + 2 a[2] + a[3]]}, {[a[2]]}]}, {[-1 + a[2]]}]}, {[a[2]]}] = , [{[{[a[1] + 2 a[2] + a[3] - 2]}, {[1 + a[2]]}]}, {[a[2]]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[5](0, 0, 0) + [1]}] = , [{}, {[0]}] = , [-1] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 1, 1)}, {f[5](1, 0, 0) + [1]}] = , [{[{[-1]}, {[0]}]}, {[1/2]}] = , [0] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1], a[4] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 0, 1, 1)}, {f[5](a[1], 0, 0) + [1]}] = , [{[{[a[1] - 3/2]}, {[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}]}, {[{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[1]}]}] = , [{[{[a[1] - 2]}, {[2]}]}, {[1]}] For the domain, {a[3] = 0, 0 <= a[1], a[4] = 1, 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], -1 + a[2], 1, 1)}, {f[5](a[1], a[2], 0) + [1]}] = , [{[{[a[1] + 2 a[2] - 2]}, {[a[2]]}]}, {[1/2 + a[2]]}] = , [a[2]] For the domain, {a[2] = 0, a[1] = 1, a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] + 1, 1), f[3](1, 0, a[3] - 1, 2)}, {f[5](1, 0, a[3]) + [1]}] = , [{[a[3]], [{[a[3] - 1]}, {[0]}]}, {[1]}] = , [{[a[3]]}, {[1]}] For the domain, {a[2] = 0, 2 <= a[1], a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 0, a[3] + 1, 1), f[3](a[1], 0, a[3] - 1, 2)}, {f[5](a[1], 0, a[3]) + [1]}] = , [{[a[1] + a[3] - 1], [{[a[1] - 3/2 + a[3]]}, {[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}]}, {[1]}] = , [{[a[1] + a[3] - 1]}, {[1]}] For the domain, {a[1] = 0, a[2] = 0, a[4] = 1, 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, a[3] - 1, 2)}, {f[5](0, 0, a[3]) + [1]}] = , [{[a[3] - 1]}, {[1]}] For the domain, {0 <= a[1], a[4] = 1, 1 <= a[2], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[3](a[1], -1 + a[2], a[3] + 1, 1), f[3](a[1], a[2], a[3] - 1, 2)}, {f[5](a[1], a[2], a[3]) + [1]}] = , [{[a[1] + 2 a[2] + a[3] - 1], [{[a[1] + 2 a[2] + a[3] - 2]}, {[a[2]]}]}, {[1 + a[2]]}] = , [{[a[1] + 2 a[2] + a[3] - 1]}, {[1 + a[2]]}] For the domain, {a[2] = 0, a[1] = 1, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] + 1, a[4]), f[3](1, 0, a[3] - 1, a[4] + 1)}, {}] = , [{[a[3]]}, {}] = , [a[3] + 1] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 1, a[4])}, {}] = , [{[0]}, {}] = , [1] For the domain, {a[2] = 0, a[1] = 1, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](1, 0, a[3] - 1, a[4] + 1), f[2](0, 0, a[3] + 1, a[4])}, {}] = , [{[a[3]]}, {}] = , [a[3] + 1] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 1, a[4])}, {}] = , [{[0]}, {}] = , [1] For the domain, {a[2] = 0, 2 <= a[1], 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1] - 1, 0, a[3] + 1, a[4]), f[3](a[1], 0, a[3] - 1, a[4] + 1)}, {}] = , [{[a[1] + a[3] - 1], [a[1] - 1/2 + a[3]]}, {}] = , [a[1] + a[3]] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1], 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1] - 1, 0, 1, a[4])}, {}] = , [{[a[1] - 1/2]}, {}] = , [a[1]] For the domain, {a[1] = 0, a[2] = 0, 2 <= a[4], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, a[3] - 1, a[4] + 1)}, {}] = , [{[a[3] - 1]}, {}] = , [a[3]] For the domain, {a[1] = 0, a[2] = 0, a[3] = 0, 2 <= a[4]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {}] = , [{}, {}] = , [0] For the domain, {0 <= a[1], 2 <= a[4], 1 <= a[2], 1 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{ f[3](a[1], -1 + a[2], a[3] + 1, a[4]), f[3](a[1], a[2], a[3] - 1, a[4] + 1) }, {}] = , [{[a[1] + 2 a[2] + a[3] - 1]}, {}] = , [a[1] + 2 a[2] + a[3]] For the domain, {a[3] = 0, 0 <= a[1], 2 <= a[4], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], -1 + a[2], 1, a[4])}, {}] = , [{[a[1] + 2 a[2] - 1]}, {}] = , [a[1] + 2 a[2]] ########## #, f[4], # ########## For the domain, {a[1] = 0, a[2] = 0, a[3] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[5](0, 0, 0)}] = , [{}, {[-1]}] = , [-2] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](0, 1, 0)}, {f[5](1, 0, 0)}] = , [{[{[-1]}, {[1/2]}]}, {[-1/2]}] = , [{[0]}, {[-1/2]}] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 1, 0)}, {f[5](a[1], 0, 0)}] = , [{[{[a[1] - 2]}, {[1/2]}]}, {[{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}]}] For the domain, {a[1] = 0, a[2] = 0, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 0, 0) + [0]}, {f[5](0, 0, 1)}] = , [{[-3]}, {[0]}] = , [-1] For the domain, {a[3] = 2, a[1] = 0, a[2] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, 1, 0) + [0]}, {f[5](0, 0, 2)}] = , [{[-3/2]}, {[0]}] = , [-1] For the domain, {a[1] = 0, a[2] = 0, 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 0, a[3] - 1, 0) + [0]}, {f[5](0, 0, a[3])}] = , [{[{[{[a[3] - 4]}, {[0]}]}, {[-1]}]}, {[0]}] = , [-1] For the domain, {a[2] = 0, a[1] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 0, 0, 0) + [0], f[4](0, 1, 1)}, {f[5](1, 0, 1)}] = , [{[{[-1]}, {[-1]}], [{[-1]}, {[1]}]}, {[0]}] = , [{[0]}, {[0]}] For the domain, {a[2] = 0, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 1, 1), f[2](a[1], 0, 0, 0) + [0]}, {f[5](a[1], 0, 1)}] = , [{[0], [{[a[1] - 2]}, {[1]}]}, {[0]}] = , [{[{[a[1] - 2]}, {[1]}]}, {[0]}] For the domain, {a[3] = 2, a[2] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 0, 1, 0) + [0], f[4](0, 1, 2)}, {f[5](1, 0, 2)}] = , [{[0], [{[-1]}, {[1]}]}, {[0]}] = , [{[0]}, {[0]}] For the domain, {a[3] = 2, a[2] = 0, a[1] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](2, 0, 1, 0) + [0], f[4](1, 1, 2)}, {f[5](2, 0, 2)}] = , [{[{[0]}, {[1]}], [{[1]}, {[{[1/2]}, {[0]}]}]}, {[0]}] = , [{[1/2]}, {[0]}] For the domain, {a[3] = 2, a[2] = 0, 3 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1], 0, 1, 0) + [0], f[4](a[1] - 1, 1, 2)}, {f[5](a[1], 0, 2)}] = , [{[{[1]}, {[1]}], [{[a[1] - 2]}, {[1]}]}, {[0]}] = , [{[{[a[1] - 2]}, {[1]}]}, {[0]}] For the domain, {a[2] = 0, 1 <= a[1], 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[4](a[1] - 1, 1, a[3]), f[2](a[1], 0, a[3] - 1, 0) + [0]}, {f[5](a[1], 0, a[3])}] = , [{[{[a[1] - 2]}, {[1]}], [ {[{[a[1] + a[3] - 7/2]}, {[{[{[a[1] - 1]}, {[2]}]}, {[1]}]}]}, {[{[{[a[1] - 2]}, {[1]}]}, {[0]}]}]}, {[0]}] = , [{[{[a[1] - 2]}, {[1]}]}, {[0]}] For the domain, {a[3] = 0, a[2] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 2, 0)}, {f[5](a[1], 1, 0)}] = , [{[a[1] - 1]}, {[1/2]}] For the domain, {a[1] = 0, a[3] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[5](0, 1, 0)}] = , [{}, {[1/2]}] = , [0] For the domain, {a[1] = 1, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 1, 0, 0) + [0], f[4](0, 2, 1)}, {f[5](1, 1, 1)}] = , [{[0], [{[0]}, {[0]}]}, {[1]}] = , [1/2] For the domain, {a[2] = 1, 2 <= a[1], a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1], 1, 0, 0) + [0], f[4](a[1] - 1, 2, 1)}, {f[5](a[1], 1, 1)}] = , [{[a[1] - 1], [{[a[1] - 3/2]}, {[0]}]}, {[1]}] = , [{[a[1] - 1]}, {[1]}] For the domain, {a[1] = 1, a[2] = 1, a[3] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](1, 1, 1, 0) + [0], f[4](0, 2, 2)}, {f[5](1, 1, 2)}] = , [{[0], [{[1/2]}, {[1/2]}]}, {[1]}] = , [1/2] For the domain, {a[2] = 1, 2 <= a[1], a[3] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 2, 2), f[2](a[1], 1, 1, 0) + [0]}, {f[5](a[1], 1, 2)}] = , [{[a[1] - 1], [{[{[a[1] - 1]}, {[a[1] - 1]}]}, {[{[1]}, {[1]}]}]}, {[1]}] = , [{[a[1] - 1]}, {[1]}] For the domain, {3 <= a[3], a[2] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[2](a[1], 1, a[3] - 1, 0) + [0], f[4](a[1] - 1, 2, a[3])}, {f[5](a[1], 1, a[3])}] = , [{[a[1] - 1], [ {[{[a[1] + a[3] - 3]}, {[{[a[1]]}, {[2]}]}]}, {[2], [{[a[1] - 1]}, {[1]}]}] }, {[1]}] = , [{[a[1] - 1]}, {[1]}] For the domain, {a[1] = 0, a[2] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 0, 0) + [0]}, {f[5](0, 1, 1)}] = , [{[-1]}, {[1]}] = , [0] For the domain, {a[1] = 0, a[2] = 1, a[3] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, 1, 0) + [0]}, {f[5](0, 1, 2)}] = , [{[{[0]}, {[0]}]}, {[1]}] = , [0] For the domain, {3 <= a[3], a[1] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 1, a[3] - 1, 0) + [0]}, {f[5](0, 1, a[3])}] = , [{[{[{[a[3] - 3]}, {[{[0]}, {[2]}]}]}, {[2], [{[-1]}, {[1]}]}]}, {[1]}] = , [0] For the domain, {a[2] = 2, a[3] = 1, 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 3, 1), f[2](a[1], 2, 0, 0) + [0]}, {}] = , [{[a[1] - 1], [a[1] - 1/2]}, {}] = , [a[1]] For the domain, {a[3] = 1, 1 <= a[1], a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1], 3, 0, 0) + [0], f[4](a[1] - 1, 4, 1)}, {}] = , [{[a[1] - 1], [{[{[{[2 a[1]]}, {[{[a[1]]}, {[{[a[1]]}, {[2]}]}]}]}, {[a[1]]}]}, {[a[1]]}] }, {}] = , [a[1]] For the domain, {a[3] = 1, 1 <= a[1], 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](a[1], a[2], 0, 0) + [0], f[4](a[1] - 1, 1 + a[2], 1)}, {}] = , [{[a[1] - 1], [{[{[{[2 a[1] + a[2] - 3]}, {[a[1] + 1/2]}]}, {[a[1]]}]}, {[a[1]]}]}, {}] = , [a[1]] For the domain, {2 <= a[2], 2 <= a[3], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 1 + a[2], a[3]), f[2](a[1], a[2], a[3] - 1, 0) + [0]}, {}] = , [{[a[1] - 1], [{[{[a[1] + a[3] - 3]}, {[1 + a[1]]}]}, {[a[1]]}]}, {}] = , [a[1]] For the domain, {a[3] = 0, 2 <= a[2], 1 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 1 + a[2], 0)}, {}] = , [{[a[1] - 1]}, {}] = , [a[1]] For the domain, {a[1] = 0, a[2] = 2, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 2, 0, 0) + [0]}, {}] = , [{[-1/4]}, {}] = , [0] For the domain, {a[1] = 0, a[3] = 1, a[2] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, 3, 0, 0) + [0]}, {}] = , [{[{[{[{[0]}, {[{[0]}, {[{[0]}, {[2]}]}]}]}, {[0]}]}, {[0]}]}, {}] = , [0] For the domain, {a[1] = 0, a[3] = 1, 4 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, a[2], 0, 0) + [0]}, {}] = , [{[{[{[{[a[2] - 3]}, {[1/2]}]}, {[0]}]}, {[0]}]}, {}] = , [0] For the domain, {a[1] = 0, 2 <= a[2], 2 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[2](0, a[2], a[3] - 1, 0) + [0]}, {}] = , [{[{[{[a[3] - 3]}, {[1]}]}, {[0]}]}, {}] = , [0] For the domain, {a[1] = 0, a[3] = 0, 2 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {}] = , [{}, {}] = , [0] ########## #, f[5], # ########## For the domain, {a[1] = 0, a[2] = 0, a[3] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {f[6](0, 0) + [0]}] = , [{}, {[0]}] = , [-1] For the domain, {a[2] = 0, a[3] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](0, 0, 1)}, {f[6](1, 0) + [0]}] = , [{[-1]}, {[0]}] = , [-1/2] For the domain, {a[2] = 0, a[3] = 0, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 0, 1)}, {f[6](a[1], 0) + [0]}] = , [{[{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[0]}] For the domain, {a[3] = 0, 0 <= a[1], 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[5](a[1], -1 + a[2], 1)}, {f[6](a[1], a[2]) + [0]}] = , [{[-1 + a[2]]}, {[a[2]]}] = , [- 1/2 + a[2]] For the domain, {a[2] = 0, a[1] = 1, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](0, 0, 2), f[3](1, 0, 0, 0) + [0]}, {f[6](1, 0) + [2]}] = , [{[-1]}, {[2]}] = , [0] For the domain, {a[2] = 0, a[1] = 2, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](1, 0, 2), f[3](2, 0, 0, 0) + [0]}, {f[6](2, 0) + [2]}] = , [{[{[0]}, {[0]}], [{[{[-1]}, {[1]}]}, {[0]}]}, {[2]}] = , [0] For the domain, {a[2] = 0, a[3] = 1, a[1] = 3} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](3, 0, 0, 0) + [0], f[4](2, 0, 2)}, {f[6](3, 0) + [2]}] = , [{[{[1/2]}, {[0]}], [{[{[0]}, {[1]}]}, {[0]}]}, {[2]}] = , [0] For the domain, {a[2] = 0, a[3] = 1, 4 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], 0, 0, 0) + [0], f[4](a[1] - 1, 0, 2)}, {f[6](a[1], 0) + [2]}] = , [{[{[{[1]}, {[1]}]}, {[0]}], [{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {[2]}] = , [0] For the domain, {a[1] = 0, a[2] = 0, a[3] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, 0, 0) + [0]}, {f[6](0, 0) + [2]}] = , [{[-2]}, {[2]}] = , [0] For the domain, {a[3] = 2, a[2] = 0, a[1] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](0, 0, 3), f[3](1, 0, 1, 0) + [0]}, {}] = , [{[-1], [{[0]}, {[0]}]}, {}] = , [0] For the domain, {a[2] = 0, a[1] = 1, 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](1, 0, a[3] - 1, 0) + [0], f[4](0, 0, a[3] + 1)}, {}] = , [{[-1], [{[{[a[3] - 2]}, {[1]}]}, {[0]}]}, {}] = , [0] For the domain, {a[3] = 2, a[2] = 0, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[4](a[1] - 1, 0, 3), f[3](a[1], 0, 1, 0) + [0]}, {}] = , [{[{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[0]}], [{[{[a[1] - 3]}, {[1]}]}, {[0]}]}, {}] = , [0] For the domain, {a[2] = 0, 2 <= a[1], 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], 0, a[3] - 1, 0) + [0], f[4](a[1] - 1, 0, a[3] + 1)}, {}] = , [ {[{[{[a[1] - 3]}, {[1]}]}, {[0]}], [{[{[a[1] - 3 + a[3]]}, {[1]}]}, {[0]}]} , {}] = , [0] For the domain, {a[3] = 2, a[1] = 0, a[2] = 0} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, 1, 0) + [0]}, {}] = , [{[-1/2]}, {}] = , [0] For the domain, {a[1] = 0, a[2] = 0, 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](0, 0, a[3] - 1, 0) + [0]}, {}] = , [{[{[{[a[3] - 3]}, {[1]}]}, {[0]}]}, {}] = , [0] For the domain, {0 <= a[1], a[3] = 1, 1 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[3](a[1], a[2], 0, 0) + [0], f[5](a[1], -1 + a[2], 2)}, {f[6](a[1], a[2]) + [2]}] = , [{[-1 + a[2]]}, {[a[2] + 2]}] = , [a[2]] For the domain, {0 <= a[1], 1 <= a[2], a[3] = 2} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[3](a[1], a[2], 1, 0) + [0], f[5](a[1], -1 + a[2], 3)}, {}] = , [{[-1 + a[2]], [{[a[2]]}, {[a[2]]}]}, {}] = , [a[2]] For the domain, {0 <= a[1], 1 <= a[2], 3 <= a[3]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [ {f[5](a[1], -1 + a[2], a[3] + 1), f[3](a[1], a[2], a[3] - 1, 0) + [0]}, {}] = , [ {[-1 + a[2]], [{[{[a[1] + 2 a[2] + a[3] - 3]}, {[1 + a[2]]}]}, {[a[2]]}]}, {}] = , [a[2]] ########## #, f[6], # ########## For the domain, {a[1] = 0, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[5](0, 0, 0) + [1]}, {}] = , [{[0]}, {}] = , [1] For the domain, {a[1] = 1, a[2] = 1} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[5](1, 0, 0) + [1]}, {}] = , [{[1/2]}, {}] = , [1] For the domain, {a[2] = 1, 2 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[5](a[1], 0, 0) + [1]}, {}] = , [{[{[{[{[a[1] - 2]}, {[2]}]}, {[1]}]}, {[1]}]}, {}] = , [1] For the domain, {0 <= a[1], 2 <= a[2]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{f[5](a[1], -1 + a[2], 0) + [1]}, {}] = , [{[- 1/2 + a[2]]}, {}] = , [a[2]] For the domain, {a[2] = 0, 0 <= a[1]} using the inductive hypothesis, and then simplifying, we get: The value of the game = , [{}, {}] = , [{}, {}] = , [0] ### The theorems are proved! ### This took, 201.067, seconds of CPU time