author  nipkow 
Wed, 04 Aug 2004 11:25:08 +0200  
changeset 15106  e8cef6993701 
parent 4276  a770eae2cdb0 
permissions  rwrr 
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%%%%Currently UNDOCUMENTED lowlevel functions! from previous manual 
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%%%%Low level information about terms and module Logic. 

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%%%%Mainly used for implementation of Pure. 

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%move to ML sources? 

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\subsection{Basic declarations} 

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The implication symbol is {\tt implies}. 

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The term \verball T is the universal quantifier for type {\tt T}\@. 

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The term \verbequals T is the equality predicate for type {\tt T}\@. 

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There are a number of basic functions on terms and types. 

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\index{>} 

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\beginprog 

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op > : typ list * typ > typ 

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\endprog 

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Given types \([ \tau_1, \ldots, \tau_n]\) and \(\tau\), it 

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forms the type \(\tau_1\to \cdots \to (\tau_n\to\tau)\). 

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Calling {\prog{}type_of \${t}}\index{type_of} computes the type of the 

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term~$t$. Raises exception {\tt TYPE} unless applications are welltyped. 

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Calling \verbsubst_bounds$([u_{n1},\ldots,u_0],\,t)$\index{subst_bounds} 

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substitutes the $u_i$ for loose bound variables in $t$. This achieves 

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\(\beta\)reduction of \(u_{n1} \cdots u_0\) into $t$, replacing {\tt 

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Bound~i} with $u_i$. For \((\lambda x y.t)(u,v)\), the bound variable 

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indices in $t$ are $x:1$ and $y:0$. The appropriate call is 

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\verbsubst_bounds([v,u],t). Loose bound variables $\geq n$ are reduced 

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by $n$ to compensate for the disappearance of $n$ lambdas. 

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\index{maxidx_of_term} 

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\beginprog 

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maxidx_of_term: term > int 

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\endprog 

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Computes the maximum index of all the {\tt Var}s in a term. 

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If there are no {\tt Var}s, the result is \(1\). 

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\index{term_match} 

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\beginprog 

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term_match: (term*term)list * term*term > (term*term)list 

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\endprog 

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Calling \verbterm_match(vts,t,u) instantiates {\tt Var}s in {\tt t} to 

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match it with {\tt u}. The resulting list of variable/term pairs extends 

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{\tt vts}, which is typically empty. Firstorder pattern matching is used 

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to implement metalevel rewriting. 

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\subsection{The representation of objectrules} 

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The module {\tt Logic} contains operations concerned with inference  

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especially, for constructing and destructing terms that represent 

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objectrules. 

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\index{occs} 

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\beginprog 

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op occs: term*term > bool 

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\endprog 

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Does one term occur in the other? 

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(This is a reflexive relation.) 

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\index{add_term_vars} 

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\beginprog 

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add_term_vars: term*term list > term list 

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\endprog 

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Accumulates the {\tt Var}s in the term, suppressing duplicates. 

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The second argument should be the list of {\tt Var}s found so far. 

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\index{add_term_frees} 

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\beginprog 

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add_term_frees: term*term list > term list 

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\endprog 

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Accumulates the {\tt Free}s in the term, suppressing duplicates. 

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The second argument should be the list of {\tt Free}s found so far. 

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\index{mk_equals} 

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\beginprog 

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mk_equals: term*term > term 

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\endprog 

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Given $t$ and $u$ makes the term $t\equiv u$. 

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\index{dest_equals} 

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\beginprog 

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dest_equals: term > term*term 

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\endprog 

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Given $t\equiv u$ returns the pair $(t,u)$. 

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\index{list_implies:} 

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\beginprog 

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list_implies: term list * term > term 

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\endprog 

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Given the pair $([\phi_1,\ldots, \phi_m], \phi)$ 

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makes the term \([\phi_1;\ldots; \phi_m] \Imp \phi\). 

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\index{strip_imp_prems} 

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\beginprog 

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strip_imp_prems: term > term list 

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\endprog 

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Given \([\phi_1;\ldots; \phi_m] \Imp \phi\) 

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returns the list \([\phi_1,\ldots, \phi_m]\). 

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\index{strip_imp_concl} 

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\beginprog 

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strip_imp_concl: term > term 

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\endprog 

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Given \([\phi_1;\ldots; \phi_m] \Imp \phi\) 

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returns the term \(\phi\). 

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\index{list_equals} 

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\beginprog 

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list_equals: (term*term)list * term > term 

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\endprog 

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For adding flexflex constraints to an objectrule. 

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Given $([(t_1,u_1),\ldots, (t_k,u_k)], \phi)$, 

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makes the term \([t_1\equiv u_1;\ldots; t_k\equiv u_k]\Imp \phi\). 

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\index{strip_equals} 

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\beginprog 

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strip_equals: term > (term*term) list * term 

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\endprog 

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Given \([t_1\equiv u_1;\ldots; t_k\equiv u_k]\Imp \phi\), 

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returns $([(t_1,u_1),\ldots, (t_k,u_k)], \phi)$. 

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\index{rule_of} 

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\beginprog 

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rule_of: (term*term)list * term list * term > term 

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\endprog 

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Makes an objectrule: given the triple 

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\[ ([(t_1,u_1),\ldots, (t_k,u_k)], [\phi_1,\ldots, \phi_m], \phi) \] 

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returns the term 

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\([t_1\equiv u_1;\ldots; t_k\equiv u_k; \phi_1;\ldots; \phi_m]\Imp \phi\) 

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\index{strip_horn} 

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\beginprog 

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strip_horn: term > (term*term)list * term list * term 

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\endprog 

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Breaks an objectrule into its parts: given 

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\[ [t_1\equiv u_1;\ldots; t_k\equiv u_k; \phi_1;\ldots; \phi_m] \Imp \phi \] 

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returns the triple 

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\(([(t_k,u_k),\ldots, (t_1,u_1)], [\phi_1,\ldots, \phi_m], \phi).\) 

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\index{strip_assums} 

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\beginprog 

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strip_assums: term > (term*int) list * (string*typ) list * term 

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\endprog 

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Strips premises of a rule allowing a more general form, 

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where $\Forall$ and $\Imp$ may be intermixed. 

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This is typical of assumptions of a subgoal in natural deduction. 

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Returns additional information about the number, names, 

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and types of quantified variables. 

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\index{strip_prems} 

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\beginprog 

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strip_prems: int * term list * term > term list * term 

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\endprog 

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For finding premise (or subgoal) $i$: given the triple 

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\( (i, [], \phi_1;\ldots \phi_i\Imp \phi) \) 

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it returns another triple, 

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\((\phi_i, [\phi_{i1},\ldots, \phi_1], \phi)\), 

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where $\phi$ need not be atomic. Raises an exception if $i$ is out of 
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range. 
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\subsection{Environments} 

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The module {\tt Envir} (which is normally closed) 

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declares a type of environments. 

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An environment holds variable assignments 

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and the next index to use when generating a variable. 

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\par\indent\vbox{\small \begin{verbatim} 

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datatype env = Envir of {asol: term xolist, maxidx: int} 

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\end{verbatim}} 

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The operations of lookup, update, and generation of variables 

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are used during unification. 

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\beginprog 

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empty: int>env 

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\endprog 

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Creates the environment with no assignments 

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and the given index. 

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\beginprog 

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lookup: env * indexname > term option 

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\endprog 

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Looks up a variable, specified by its indexname, 

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and returns {\tt None} or {\tt Some} as appropriate. 

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\beginprog 

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update: (indexname * term) * env > env 

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\endprog 

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Given a variable, term, and environment, 

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produces {\em a new environment\/} where the variable has been updated. 

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This has no side effect on the given environment. 

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\beginprog 

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genvar: env * typ > env * term 

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\endprog 

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Generates a variable of the given type and returns it, 

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paired with a new environment (with incremented {\tt maxidx} field). 

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\beginprog 

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alist_of: env > (indexname * term) list 

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\endprog 

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Converts an environment into an association list 

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containing the assignments. 

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\beginprog 

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norm_term: env > term > term 

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\endprog 

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Copies a term, 

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following assignments in the environment, 

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and performing all possible \(\beta\)reductions. 

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\beginprog 

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rewrite: (env * (term*term)list) > term > term 

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\endprog 

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Rewrites a term using the given term pairs as rewrite rules. Assignments 

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are ignored; the environment is used only with {\tt genvar}, to generate 

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unique {\tt Var}s as placeholders for bound variables. 

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\subsection{The unification functions} 

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\beginprog 

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unifiers: env * ((term*term)list) > (env * (term*term)list) Seq.seq 
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\endprog 
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This is the main unification function. 

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Given an environment and a list of disagreement pairs, 

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it returns a sequence of outcomes. 

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Each outcome consists of an updated environment and 

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a list of flexflex pairs (these are discussed below). 

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\beginprog 

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smash_unifiers: env * (term*term)list > env Seq.seq 
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\endprog 
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This unification function maps an environment and a list of disagreement 

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pairs to a sequence of updated environments. The function obliterates 
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flexflex pairs by choosing the obvious unifier. It may be used to tidy up 
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any flexflex pairs remaining at the end of a proof. 
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\subsubsection{Multiple unifiers} 

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The unification procedure performs Huet's {\sc match} operation 

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\cite{huet75} in big steps. 

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It solves \(\Var{f}(t_1,\ldots,t_p) \equiv u\) for \(\Var{f}\) by finding 

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all ways of copying \(u\), first trying projection on the arguments 

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\(t_i\). It never copies below any variable in \(u\); instead it returns a 

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new variable, resulting in a flexflex disagreement pair. 

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\beginprog 

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type_assign: cterm > cterm 

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\endprog 

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Produces a cterm by updating the signature of its argument 

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to include all variable/type assignments. 

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Type inference under the resulting signature will assume the 

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same type assignments as in the argument. 

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This is used in the goal package to give persistence to type assignments 

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within each proof. 

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(Contrast with {\sc lcf}'s sticky types \cite[page 148]{paulsonbook}.) 

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