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Transformation rules are used to transform the given formula into an equivalent formula that simplifies the inductive part of the proof. For example, the only logical symbols in the transformed formula are ¬ {\displaystyle \neg } , ∧ {\displaystyle \land } , and ∃ {\displaystyle \exists } , so the induction handles logical symbols with ...
If instead the rules are just {}, then we obtain a presentation of the bicyclic monoid. Thus semi-Thue systems constitute a natural framework for solving the word problem for monoids and groups. In fact, every monoid has a presentation of the form ( Σ , R ) {\displaystyle (\Sigma ,R)} , i.e. it may always be presented by a semi-Thue system ...
In logic, a rule of replacement [1] [2] [3] is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a ...
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs.
Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependence, i.e., = (,) and = (,).The functional form of Hamilton's equations is ˙ =, ˙ = In general, a transformation (q, p) → (Q, P) does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some ...
Transformation into negation normal form can increase the size of a formula only linearly: the number of occurrences of atomic formulas remains the same, the total number of occurrences of and is unchanged, and the number of occurrences of in the normal form is bounded by the length of the original formula.
The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = , sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: = +, = +, = +.
Every first-order formula is logically equivalent (in classical logic) to some formula in prenex normal form. [3] There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connectives appear in the formula.