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| '''HOL Light''' is a member of the [[HOL theorem prover family]]. Like the other members, it is a [[proof assistant]] for classical [[higher order logic]]. Compared with other HOL systems, HOL Light is intended to have relatively simple foundations. HOL Light is authored and maintained by the mathematician and computer scientist [[John Harrison (mathematician)|John Harrison]]. HOL Light is released under the [[BSD licenses#2-clause|simplified BSD license]].<ref>http://code.google.com/p/hol-light/</ref>
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| ==Logical foundations==
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| HOL Light is based on a formulation of [[type theory]] with equality
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| as the only [[primitive notion]]. The primitive rules of inference
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| are the following:
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| {| class="wikitable"
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| |-
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| | style="text-align: center;" | <math> \cfrac{\qquad }{ \vdash t = t}</math>
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| | style="text-align: center;" | REFL
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| | reflexivity of equality
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma \vdash s = t \qquad \Delta \vdash t = u}
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| {\Gamma \cup \Delta \vdash s = u}
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| </math>
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| | style="text-align: center;" | TRANS
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| | transitivity of equality
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma \vdash f = g \qquad \Delta \vdash x = y}
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| {\Gamma \cup \Delta \vdash f(x) = g(y)}
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| </math>
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| | style="text-align: center;" | MK_COMB
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| | congruence of equality
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma \vdash s = t}{\Gamma \vdash (\lambda x. s) = (\lambda x. t)}
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| </math>
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| | style="text-align: center;" | ABS
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| | abstraction of equality (<math>x</math> must not be free in <math>\Gamma</math>)
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| |-
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| | style="text-align: center;" | <math>\cfrac{\qquad}{\vdash (\lambda x. t) x = t}
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| </math>
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| | style="text-align: center;" | BETA
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| | connection of abstraction and function application
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| |-
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| | style="text-align: center;" | <math> \cfrac{\qquad }{ \{p\} \vdash p}
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| </math>
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| | style="text-align: center;" | ASSUME
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| | assuming <math>p</math>, prove <math>p</math>
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma \vdash p = q \qquad \Delta \vdash p}
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| {\Gamma \cup \Delta \vdash q}
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| </math>
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| | style="text-align: center;" | EQ_MP
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| | relation of equality and deduction
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma \vdash p \qquad \Delta \vdash q}
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| {(\Gamma - \{q\}) \cup (\Delta - \{p\}) \vdash p = q}
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| </math>
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| | style="text-align: center;" | DEDUCT_ANTISYM_RULE
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| | deduce equality from 2-way deducibility
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma[x_1,\ldots,x_n] \vdash p[x_1,\ldots,x_n]}
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| {\Gamma[t_1,\ldots,t_n] \vdash p[t_1,\ldots,t_n]}
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| </math> | |
| | style="text-align: center;" | INST
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| | instantiate variables in assumptions and conclusion of theorem
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| |-
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| | style="text-align: center;" | <math> \cfrac{\Gamma[\alpha_1,\ldots,\alpha_n] \vdash p[\alpha_1,\ldots,\alpha_n]}
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| {\Gamma[\tau_1,\ldots,\tau_n] \vdash p[\tau_1,\ldots,\tau_n]}
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| </math>
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| | style="text-align: center;" | INST_TYPE
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| | instantiate type variables in assumptions and conclusion of theorem
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| |}
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| This formulation of type theory is very close to the one described in
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| section II.2 of {{Harvtxt|Lambek|Scott|1986}}.
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| ==References==
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| {{reflist}}
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| *{{Citation
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| | last = Lambek
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| | first = J
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| | coauthors = P. J. Scott
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| | title = Introduction to Higher Order Categorical logic
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| | publisher = Cambridge University Press
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| | year = 1986
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| }}
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| ==Further reading==
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| *{{Citation
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| | author = Freek Wiedijk
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| | title = Formal Proof — Getting Started
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| | journal = [[Notices of the American Mathematical Society]]
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| |date=December 2008
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| | volume = 55
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| | issue = 11
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| | pages = 1408–1414
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| | url = http://www.ams.org/notices/200811/tx081101408p.pdf
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| | accessdate = 2008-12-14
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| }}
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| ==External links==
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| * [http://www.cl.cam.ac.uk/users/jrh/hol-light/ HOL Light]
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| [[Category:Free theorem provers]]
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| [[Category:Proof assistants]]
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| [[Category:OCaml software]]
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