diff -r f614c619b1e1 -r a54ca4e90874 doc-src/IsarImplementation/Thy/document/logic.tex --- a/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 21:42:21 2006 +0200 +++ b/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 22:48:37 2006 +0200 @@ -493,7 +493,7 @@ \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ f\ x\ {\isasymequiv}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isasymequiv}\ g} & extensionality \\ \isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & logical equivalence \\ \end{tabular} - \caption{Conceptual axiomatization of \isa{{\isasymequiv}}}\label{fig:pure-equality} + \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality} \end{center} \end{figure} @@ -512,10 +512,6 @@ \cite{Barendregt-Geuvers:2001}, where \isa{x\ {\isacharcolon}\ A} hypotheses are treated explicitly for types, in the same way as propositions.} - \medskip FIXME \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence and primitive definitions - - Since the basic representation of terms already accounts for \isa{{\isasymalpha}}-conversion, Pure equality essentially acts like \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence on terms, while coinciding with bi-implication. - \medskip The axiomatization of a theory is implicitly closed by forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} holds for any substitution instance of an axiom \isa{{\isasymturnstile}\ A}. By pushing substitution through derivations @@ -544,7 +540,32 @@ In principle, variables could be substituted in hypotheses as well, but this would disrupt monotonicity reasoning: deriving \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymturnstile}\ B{\isasymvartheta}} from \isa{{\isasymGamma}\ {\isasymturnstile}\ B} is correct, but \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymsupseteq}\ {\isasymGamma}} does not necessarily hold --- the result - belongs to a different proof context.% + belongs to a different proof context. + + \medskip The system allows axioms to be stated either as plain + propositions, or as arbitrary functions (``oracles'') that produce + propositions depending on given arguments. It is possible to trace + the used of axioms (and defined theorems) in derivations. + Invocations of oracles are recorded invariable. + + Axiomatizations should be limited to the bare minimum, typically as + part of the initial logical basis of an object-logic formalization. + Normally, theories will be developed definitionally, by stating + restricted equalities over constants. + + A \emph{simple definition} consists of a constant declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} together with an axiom \isa{{\isasymturnstile}\ c\ {\isasymequiv}\ t}, where \isa{t} is a closed term without any hidden polymorphism. The RHS may + depend on further defined constants, but not \isa{c} itself. + Definitions of constants with function type may be presented + liberally as \isa{c\ \isactrlvec \ {\isasymequiv}\ t} instead of the puristic \isa{c\ {\isasymequiv}\ {\isasymlambda}\isactrlvec x{\isachardot}\ t}. + + An \emph{overloaded definition} consists may give zero or one + equality axioms \isa{c{\isacharparenleft}{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}{\isacharparenright}\ {\isasymequiv}\ t} for each type + constructor \isa{{\isasymkappa}}, with distinct variables \isa{\isactrlvec {\isasymalpha}} + as formal arguments. The RHS may mention previously defined + constants as above, or arbitrary constants \isa{d{\isacharparenleft}{\isasymalpha}\isactrlisub i{\isacharparenright}} + for some \isa{{\isasymalpha}\isactrlisub i} projected from \isa{\isactrlvec {\isasymalpha}}. + Thus overloaded definitions essentially work by primitive recursion + over the syntactic structure of a single type argument.% \end{isamarkuptext}% \isamarkuptrue% % @@ -559,15 +580,78 @@ \indexmltype{ctyp}\verb|type ctyp| \\ \indexmltype{cterm}\verb|type cterm| \\ \indexmltype{thm}\verb|type thm| \\ + \indexml{proofs}\verb|proofs: int ref| \\ + \indexml{Thm.ctyp-of}\verb|Thm.ctyp_of: theory -> typ -> ctyp| \\ + \indexml{Thm.cterm-of}\verb|Thm.cterm_of: theory -> term -> cterm| \\ + \indexml{Thm.assume}\verb|Thm.assume: cterm -> thm| \\ + \indexml{Thm.forall-intr}\verb|Thm.forall_intr: cterm -> thm -> thm| \\ + \indexml{Thm.forall-elim}\verb|Thm.forall_elim: cterm -> thm -> thm| \\ + \indexml{Thm.implies-intr}\verb|Thm.implies_intr: cterm -> thm -> thm| \\ + \indexml{Thm.implies-elim}\verb|Thm.implies_elim: thm -> thm -> thm| \\ + \indexml{Thm.generalize}\verb|Thm.generalize: string list * string list -> int -> thm -> thm| \\ + \indexml{Thm.instantiate}\verb|Thm.instantiate: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm| \\ + \indexml{Thm.get-axiom-i}\verb|Thm.get_axiom_i: theory -> string -> thm| \\ + \indexml{Thm.invoke-oracle-i}\verb|Thm.invoke_oracle_i: theory -> string -> theory * Object.T -> thm| \\ + \indexml{Theory.add-axioms-i}\verb|Theory.add_axioms_i: (string * term) list -> theory -> theory| \\ + \indexml{Theory.add-deps}\verb|Theory.add_deps: string -> string * typ -> (string * typ) list -> theory -> theory| \\ + \indexml{Theory.add-oracle}\verb|Theory.add_oracle: string * (theory * Object.T -> term) -> theory -> theory| \\ + \indexml{Theory.add-defs-i}\verb|Theory.add_defs_i: bool -> bool -> (bstring * term) list -> theory -> theory| \\ \end{mldecls} \begin{description} - \item \verb|ctyp| FIXME + \item \verb|ctyp| and \verb|cterm| represent certified types + and terms, respectively. These are abstract datatypes that + guarantee that its values have passed the full well-formedness (and + well-typedness) checks, relative to the declarations of type + constructors, constants etc. in the theory. + + This representation avoids syntactic rechecking in tight loops of + inferences. There are separate operations to decompose certified + entities (including actual theorems). + + \item \verb|thm| represents proven propositions. This is an + abstract datatype that guarantees that its values have been + constructed by basic principles of the \verb|Thm| module. + + \item \verb|proofs| determines the detail of proof recording: \verb|0| + records only oracles, \verb|1| records oracles, axioms and named + theorems, \verb|2| records full proof terms. + + \item \verb|Thm.assume|, \verb|Thm.forall_intr|, \verb|Thm.forall_elim|, \verb|Thm.implies_intr|, and \verb|Thm.implies_elim| + correspond to the primitive inferences of \figref{fig:prim-rules}. + + \item \verb|Thm.generalize|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharcomma}\ \isactrlvec x{\isacharparenright}} + corresponds to the \isa{generalize} rules of + \figref{fig:subst-rules}. A collection of type and term variables + is generalized simultaneously, according to the given basic names. - \item \verb|cterm| FIXME + \item \verb|Thm.instantiate|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}\isactrlisub s{\isacharcomma}\ \isactrlvec x\isactrlisub {\isasymtau}{\isacharparenright}} corresponds to the \isa{instantiate} rules + of \figref{fig:subst-rules}. Type variables are substituted before + term variables. Note that the types in \isa{\isactrlvec x\isactrlisub {\isasymtau}} + refer to the instantiated versions. + + \item \verb|Thm.get_axiom_i|~\isa{thy\ name} retrieves a named + axiom, cf.\ \isa{axiom} in \figref{fig:prim-rules}. - \item \verb|thm| FIXME + \item \verb|Thm.invoke_oracle_i|~\isa{thy\ name\ arg} invokes the + oracle function \isa{name} of the theory. Logically, this is + just another instance of \isa{axiom} in \figref{fig:prim-rules}, + but the system records an explicit trace of oracle invocations with + the \isa{thm} value. + + \item \verb|Theory.add_axioms_i|~\isa{{\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ A{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} adds + arbitrary axioms, without any checking of the proposition. + + \item \verb|Theory.add_oracle|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}} declares an + oracle, i.e.\ a function for generating arbitrary axioms. + + \item \verb|Theory.add_deps|~\isa{name\ c\isactrlisub {\isasymtau}\ \isactrlvec d\isactrlisub {\isasymsigma}} declares dependencies of a new specification for + constant \isa{c\isactrlisub {\isasymtau}} from relative to existing ones for + constants \isa{\isactrlvec d\isactrlisub {\isasymsigma}}. + + \item \verb|Theory.add_defs_i|~\isa{unchecked\ overloaded\ {\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ c\ \isactrlvec x\ {\isasymequiv}\ t{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} states a definitional axiom for an already + declared constant \isa{c}. Dependencies are recorded using \verb|Theory.add_deps|, unless the \isa{unchecked} option is set. \end{description}% \end{isamarkuptext}% @@ -653,7 +737,19 @@ \begin{description} - \item FIXME + \item \verb|Conjunction.intr| derives \isa{A\ {\isacharampersand}\ B} from \isa{A} and \isa{B}. + + \item \verb|Conjunction.intr| derives \isa{A} and \isa{B} + from \isa{A\ {\isacharampersand}\ B}. + + \item \verb|Drule.mk_term|~\isa{t} derives \isa{TERM\ t}. + + \item \verb|Drule.dest_term|~\isa{TERM\ t} recovers term \isa{t}. + + \item \verb|Logic.mk_type|~\isa{{\isasymtau}} produces the term \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}}. + + \item \verb|Logic.dest_type|~\isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} recovers the type + \isa{{\isasymtau}}. \end{description}% \end{isamarkuptext}%