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– Modus ponens:
Γ 1 ⊢ t 1 ⇒ t 2 Γ 2 ⊢ t 1
Γ 1 ∪ Γ 2 ⊢ t 2
The deductive system of HOL is shown to be sound for the set theoretic semantics
of HOL described in the previous section. Each mechanism of it guarantees to
preserve the property of possessing a model. Thus, theories built up from the
initial HOL theory (which does possess a model) using these mechanisms are
guaranteed to be consistent.
HOL notation Standard notation Description
Truth T ⊤ true
Falsity F ⊥ false
Negation ~t ¬t not t
Disjunction t 1\/t 2 t 1 ∨ t 2 t 1 or t 2
Conjunction t 1/\t 2 t 1 ∧ t 2 t 1 and t 2
Implication t 1==>t 2 t 1 ⇒ t 2 t 1 implies t 2
Equality t 1=t 2 t 1 = t 2 t 1 equals t 2
∀-quantification !x.t ∀x. t for all x : t
∃-quantification ?x.t ∃x. t for some x : t
ε-term @x.t εx. t an x such that: t
Table 1. Terms of the HOL Logic [6]
The theory INIT is the initial theory of the HOL logic. It introduces the
logical operators shown in Table 1 and contains the five axioms that the HOL
system is based on.
BOOL_CASES_AX |- !t. (t = T) \/ (t = F)
IMP_ANTISYM_AX |- !t1 t2. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 = t2)
ETA_AX
|- !t. (\x. t x) = t
SELECT_AX |- !P:’a->bool x. P x ==> P($@ P)
INFINITY_AX |- ?f:ind->ind. ONE_ONE f /\ ~(ONTO f)
where ONE_ONE and ONTO aredefinedasfollows
ONE_ONE_DEF |- ONE_ONE f = (!x1 x2. (f x1 = f x2) ==> (x1 = x2))
ONTO_DEF |- ONTO f = (!y. ?x. y = f x)
In HOL, a formula is represented by a then ML type term whose HOL type
is bool. It can be constructed by the given constructors. Unlike this, a theorem
(ML type thm) does not have a primitive constructor function. In this way, the
ML type asserts that theorems are not arbitrarily and unrecordedly constructed,
compromising of the consistency of the logic.
2.3 Usage
As already stated in the introduction, the HOL system provides an environment
for writing specifications and creating formal proofs of properties.