Download Automated Mathematical Induction by Francois Bronsard, Uday S. Reddy, Robert W. Hasker (auth.), PDF

By Francois Bronsard, Uday S. Reddy, Robert W. Hasker (auth.), Hantao Zhang (eds.)

It has been proven how the typical constitution that defines a kinfolk of proofs should be expressed as an evidence plan [5]. This universal constitution could be exploited within the look for specific proofs. an evidence plan has complementary parts: an explanation approach and an evidence tactic. by way of prescribing the constitution of an explanation on the point of primitive inferences, a tactic [11] presents the warrantly a part of the facts. by contrast, a style offers a extra declarative rationalization of the evidence by way of preconditions. every one process has linked results. The execution of the consequences simulates the appliance of the corresponding tactic. Theorem proving within the facts making plans framework is a two-phase strategy: 1. Tactic building is by means of a technique of procedure composition: Given a target, an appropriate strategy is chosen. The applicability of a mode is dependent upon comparing the method's preconditions. the strategy results are then used to calculate subgoals. This strategy is utilized recursively till not more subgoals stay. a result of one-to-one correspondence among equipment and strategies, the output from this technique is a composite tactic adapted to the given objective. 2. Tactic execution generates an explanation within the object-level good judgment. be aware that no seek is taken with the execution of the method. the entire seek is treated in the course of the making plans procedure. the genuine advantages of getting separate making plans and execution levels develop into appar­ ent while an evidence try fails.

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Extra resources for Automated Mathematical Induction

Example text

The use of Q(l' 01 unify(a, b), m' 0l unify(a, b)) is valid, since Q(l' 0l unify(a, b), m' 0l unify(a, b)) is smaller than Q(a ·l', b· m') by the order >--. ve· a 0t e = bOt e 1\ [' 01 e = m' 01 e ::::} e :;" unify(a, b) Os unidy-I([ 01 unify(a, b), mOl unify(a, b)): After simplification by Lemma 30, we are left to show ve . a 0t e = b 0t e 1\ [' 01 e = m' 0l e : : } v[·[ 0t e = [Ot unify(a, b) 0t unify-I(l O{ unify(a, b), m 01 unify(a, b)) 0t e (40) As shown in the previous cases, unifiable(a, b) and unifiable-I (l' o{ unify(a, b), m' 01 unify(a, b)) hold.

Cond) inCf do: a) Generate induction conclusion: Let fJ be the mgu of t and 8 = f(81, ... , 8n ). The induction conclusion is (fJ e, conde, reple) where fJ c is the restriction of fJ on Vars(t); conde = fJ( cond); rep1e = {p t--- fJ( 8)}. b) Generate the induction hypotheses: Let fJi be the mgu of t and 8' = f (8 1, ... , 8~) for some i. The ith induction hypothesis based on fJi is: (B i , cond i , rep1i) where Bi is the restriction of fJi on Vars(t); cond i = fJi(cond); rep1i = {p t--- fJi(S')}.

Recall that Rouyer and Lescanne's proof is more thorough than ours, so we included in the fourth column an estimation of the size of their proof restricted to the unification problem. Interestingly, the difference in complexity between our proof and Rouyer and Lescanne's proof seems to arise from the use of conventional induction. The correctness theorem has the form \/x: 7J . ¢(x) 1\ \/y: 72' 1jJ(y) where 7J is the type of terms, 72 is the type of lists of terms, ¢( x) expresses the correctness of the unify function for terms, and 1jJ(y) expresses the correctness of the unify function for lists of terms.

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