A Tree of Execution of a Macroinstruction

The articles [22], [14], [25], [15], [1], [20], [3], [4], [16], [26], [11], [13], [12], [5], [6], [21], [9], [8], [10], [2], [7], [18], [23], [19], [24], and [17] provide the notation and terminology for this paper. For simplicity, we adopt the following convention: x, y, X are sets, m, n are natural numbers, O is an ordinal number, and R, S are binary relations. Let D be a set, let f be a partial function from D to N, and let n be a set. One can verify that f(n) is natural. Let R be an empty binary relation and let X be a set. Observe that R↾X is empty. One can prove the following two propositions: (1) If domR = {x} and rngR = {y}, then R = x7−→ . y. (2) field{〈x, x〉} = {x}. Let X be an infinite set and let a be a set. One can verify that X 7−→ a is infinite. One can check that there exists a function which is infinite. Let R be a finite binary relation. One can verify that fieldR is finite. The following proposition is true (3) If fieldR is finite, then R is finite.