reduction.ott

defns
JmatchP :: JM_ ::=

defn
|- expr matches pattern :: :: matchP :: matchP_ {{ com Pattern matching }} by

{{ com
Determines if a value matches a pattern.
}}

------------------------------------------------------- :: var
|- v matches x

------------------------------------------------------- :: any
|- v matches _

------------------------------------------------------- :: constant
|- constant matches constant

|- v matches pat
------------------------------------------------------- :: alias
|- v matches pat as x

|- v matches pat
------------------------------------------------------- :: typed
|- v matches (pat : t)

|- v matches pat1
------------------------------------------------------- :: or_left
|- v matches pat1 | pat2

|- v matches pat2
------------------------------------------------------- :: or_right
|- v matches pat1 | pat2

|- v1 matches pat1   ...   |- vn matches patn
------------------------------------------------------- :: construct
|- constr (v1,...,vn) matches constr (pat1,...,patn)

------------------------------------------------------- :: construct_any
|- constr (v1,...,vn) matches constr _

|- v1 matches pat1   ....   |- vn matches patn
------------------------------------------------------- :: tuple
|- (v1,....,vn) matches (pat1,....,patn)

  field_name1'=v1'...field_namen'=vn' fn1=v1''..fnl=vl'' PERMUTES field_name1=v1...field_namem=vm
  |- v1' matches pat1  ...   |- vn' matches patn
  field_name1 ... field_namem distinct
 ----------------------------------------------------- :: record
  |- {field_name1=v1;...;field_namem=vm} matches {field_name1'=pat1;...;field_namen'=patn}

|- v1 matches pat1
|- v2 matches pat2
------------------------------------------------------- :: cons
|- v1::v2 matches pat1::pat2

defns
Jmatch :: JM_ ::=

defn
|- expr matches pattern gives '<<' substs_x '>>' :: :: match :: match_ {{ com Pattern matching with substitution creation }} by

{{ com
Determines if a value matches a pattern and destructures the value into a
substitution according to the pattern's variables.  The previous pattern
matching relation is used to get deterministic behavior for \texttt{|}
patterns.
}}

------------------------------------------------------- :: var
|- v matches x gives <<x<-v>>

------------------------------------------------------- :: any
|- v matches _ gives <<:substs_x_xs:>>

------------------------------------------------------- :: constant
|- constant matches constant gives <<:substs_x_xs:>>

|- v matches pat gives <<x1<-v1,..,xn<-vn>>
------------------------------------------------------- :: alias
|- v matches pat as x gives <<x1<-v1,..,xn<-vn,x<-v>>

|- v matches pat gives <<x1<-v1,..,xn<-vn>>
------------------------------------------------------- :: typed
|- v matches (pat : t) gives <<x1<-v1,..,xn<-vn>>

|- v matches pat1 gives <<x1<-v1,..,xn<-vn>>
------------------------------------------------------- :: or_left
|- v matches pat1 | pat2 gives <<x1<-v1,..,xn<-vn>>

|- v notmatches pat1
|- v matches pat2 gives <<x1<-v1,..,xn<-vn>>
------------------------------------------------------- :: or_right
|- v matches pat1 | pat2 gives <<x1<-v1,..,xn<-vn>>

|- v1 matches pat1 gives <<substs_x1>>   ...   |- vn matches patn gives <<substs_xn>>
------------------------------------------------------- :: construct
|- constr (v1,...,vn) matches constr (pat1,...,patn) gives <<substs_x1@...@substs_xn>>

------------------------------------------------------- :: construct_any
|- constr (v1,...,vn) matches constr _ gives <<:substs_x_xs:>>

|- v1 matches pat1 gives <<substs_x1>>   ....   |- vn matches patn gives <<substs_xn>>
------------------------------------------------------- :: tuple
|- (v1,....,vn) matches (pat1,....,patn) gives <<substs_x1@....@substs_xn>>

  field_name1'=v1'...field_namen'=vn' fn1=v1''..fnl=vl'' PERMUTES field_name1=v1...field_namem=vm
  |- v1' matches pat1 gives <<substs_x1>>   ...   |- vn' matches patn gives <<substs_xn>>
  field_name1 ... field_namem distinct
 ----------------------------------------------------- :: record
  |- {field_name1=v1;...;field_namem=vm} matches {field_name1'=pat1;...;field_namen'=patn} gives <<substs_x1@...@substs_xn>>

|- v1 matches pat1 gives <<substs_x1>>
|- v2 matches pat2 gives <<substs_x2>>
------------------------------------------------------- :: cons
|- v1::v2 matches pat1::pat2 gives <<substs_x1@substs_x2>>



defns
Jrecfun :: J ::=

defn
recfun ( letrec_bindings , pattern_matching ) gives expr :: :: recfun :: recfun_ {{ com Recursive function helper }} by

{{ com
Expands a recursive definition.
}}

%%WIDTH: >~a4/landscape/normalsize
letrec_bindings = x1 = function pattern_matching1 and ... and xn = function pattern_matchingn
------------------------------------------------------- :: letrec
recfun(letrec_bindings, pattern_matching) gives <<x1 <- let rec letrec_bindings in x1,...,xn <- let rec letrec_bindings in xn>> (function pattern_matching)



defns
Jfunval :: J ::=
defn
|- funval ( e ) :: :: funval :: funval_ {{ com Function values }} by

{{ com
Determines if an expression is a function value, for use in
(\ottdrulename{Jbprim\_equal\_fun}).
}}

------------------------------------------------------- :: up
|- funval((%prim unary_prim))

------------------------------------------------------- :: bp
|- funval((%prim binary_prim))

------------------------------------------------------- :: bp_app
|- funval((%prim binary_prim) v)

------------------------------------------------------- :: func
|- funval(function pattern_matching)



defns
JRuprim :: J ::=

defn
|- unary_prim expr labelled_arrow expr' :: :: Ruprim :: uprim_ {{ com Unary primitive evaluation }} by

{{ com
Computes the result of a unary primitive application.
}}

-------------------------------------------------- :: not_true
|- not true --> false

-------------------------------------------------- :: not_false
|- not false --> true

-------------------------------------------------- :: uminus
|- ~- intn --> 0 .- intn

{{ com
The effect of creating a reference is communicated to the store via
the label on the reduction arrow. Similarly the reduction arrow
carries the value read from the store when accessing a location.
}}

--------------------------------------- :: ref_alloc
|- ref v -->:Lab_paren:(ref v = l) l

----------------- :: deref
|- !l -->:Lab_paren:(!l=v) v



defns
JRbprim :: J ::=

defn
|- expr1 binary_prim expr2 labelled_arrow expr :: :: Rbprim :: bprim_ {{ com Binary primitive evaluation }} by

{{ com
Computes the result of a binary primitive application.
}}


------------------------------------------------------- :: equal_const_true
|- constant = constant --> true

constant noteq constant'
------------------------------------------------------- :: equal_const_false
|- constant = constant' --> false

------------------------------------------------------- :: equal_loc
|- l = l' --> ((%prim =) ((%prim !) l)) ((%prim !) l')

------------------------------------------------------- :: equal_cons
|- (v1::v2) = (v1'::v2') --> (((%prim =) v1) v1') && (((%prim =) v2) v2')

------------------------------------------------------- :: equal_cons_nil
|- (v1::v2) = [] --> false

------------------------------------------------------- :: equal_nil_cons
|- [] = (v1::v2) --> false

length (v1) .... (vn) >= 2
------------------------------------------------------- :: equal_tuple
|- (v1, ...., vn) = (v1', ...., vn') --> AND(((%prim =) v1) v1' && .... && ((%prim =) vn) vn')


------------------------------------------------------- :: equal_constr
|- (constr (v1, ..., vn)) = (constr (v1', ..., vn')) --> AND(((%prim =) v1) v1' && ... && ((%prim =) vn) vn')

constr noteq constr'
------------------------------------------------------- :: equal_constr_false
|- constr (v1, ..., vm) = constr' (v1', ..., vn') --> false

------------------------------------------------------- :: equal_const_constr_false
|- constr' = constr (v1, ..., vn) --> false

------------------------------------------------------- :: equal_constr_const_false
|- constr (v1, ..., vn) = constr' --> false


  v' = {fn1''=v1''; ...; fnm''=vm''}
  fn1...fnn PERMUTES fn1''...fnm''
 ----------------------------------------------------- :: equal_rec
  |- {fn1=v1; ...; fnn=vn} = v' --> AND(((%prim =) v1) (v'.fn1) && ... && ((%prim =) vn) (v'.fnn))

------------------------------------------------------- :: plus
|- intn1 + intn2 --> intn1 .+ intn2

------------------------------------------------------- :: minus
|- intn1 - intn2 --> intn1 .- intn2

------------------------------------------------------- :: times
|- intn1 * intn2 --> intn1 .* intn2

------------------------------------------------------- :: div0
|- intn / 0 --> None

intn2 noteq 0
------------------------------------------------------- :: div
|- intn1 / intn2 --> Some (intn1 ./ intn2)

{{ com
The side effect of an assignment is communicated to the store via the
label on the reduction arrow.
}}

------------------------------------------------------- :: assign
|- l := v -->:Lab_paren:(l:=v) ()

%% Note that at the moment, JRmatching_step and JRmatching_success
%% do not depend on expression reduction, so they appear
%% before reduction (but after JM_match). However, adding guards would
%% make expression reduction and matching steps mutually recursive.

defns
JRmatching_step :: JR ::=

defn
|- expr with pattern_matching --> pattern_matching' :: :: matching_step :: matching_ {{ com Pattern matching step }} by

{{ com
Proceeding to the next case because the first, but not only, case has failed to
match.
}}

|- v notmatches pat
length (e1) ... (en) >= 1
------------------------------------------------------- :: next
|- v with pat -> e | pat1->e1 | ... | patn->en --> pat1->e1 | ... | patn->en

defns
JRmatching_success :: JR ::=

defn
|- expr with pattern_matching --> expr' :: :: matching_success :: matching_ {{ com Pattern matching finished }} by

{{ com
Proceeding to an expression because the first case matches, or the only case
does not match.
}}

|- v matches pat gives <<x1<-v1,..,xm<-vm>>
------------------------------------------------------- :: found
|- v with pat -> e | pat1->e1 | ... | patn->en --> <<x1<-v1,..,xm<-vm>> e

defns
Jred :: JR_ ::=

defn
|- expr labelled_arrow expr' :: :: expr :: expr_  {{ com Expression evaluation }} by

{{ com
Reduces an expression one-step.
}}

|- unary_prim v -->L e
------------------------------------------------------- :: uprim
|- (%prim unary_prim) v -->L e

|- v1 binary_prim v2 -->L e
------------------------------------------------------- :: bprim
|- ((%prim binary_prim) v1) v2 -->L e


------------------------------------------------------- :: typed_ctx
|- (e : t) --> e

{{ com
Right-to-left evaluation order for application (i.e., argument before function).
}}

|- e0 -->L e0'
------------------------------------------------------- :: apply_ctx_arg
|- e1 e0 -->L e1 e0'

|- e1 -->L e1'
------------------------------------------------------- :: apply_ctx_fun
|- e1 v0 -->L e1' v0

------------------------------------------------------- :: apply
|- (function pattern_matching v0) --> match v0 with pattern_matching

|- e0 -->L e0'
------------------------------------------------------- :: let_ctx
|- let pat = e0 in e -->L let pat = e0' in e

|- v matches pat gives <<x1<-v1,..,xm<-vm>>
------------------------------------------------------- :: let_subst
|- let pat = v in e --> <<x1<-v1,..,xm<-vm>>e


letrec_bindings = x1 = function pattern_matching1 and ... and xn = function pattern_matchingn
recfun(letrec_bindings, pattern_matching1) gives e1 ... recfun(letrec_bindings, pattern_matchingn) gives en
------------------------------------------------------- :: letrec
|- let rec letrec_bindings in e --> <<x1 <- e1,...,xn <- en>>e

|- e1 -->L e1'
------------------------------------------------------- :: sequence_ctx_left
|- e1; e2 -->L e1'; e2

------------------------------------------------------- :: sequence
|- v; e2 --> e2

|- e1 -->L e1'
------------------------------------------------------- :: ifthenelse_ctx
|- if e1 then e2 else e3 -->L if e1' then e2 else e3

------------------------------------------------------- :: ifthenelse_true
|- if true then e2 else e3 --> e2

------------------------------------------------------- :: ifthenelse_false
|- if false then e2 else e3 --> e3

{{ com
We treat matching one pattern against one value as atomic (this would be
relevant when matching the contents of a reference after introducing concurrent
evaluation).
}}

|- e -->L e'
------------------------------------------------------- :: match_ctx
|- match e with pattern_matching -->L match e' with pattern_matching

|- v with pattern_matching --> pattern_matching'
------------------------------------------------------- :: match_step
|- match v with pattern_matching --> match v with pattern_matching'

|- v with pattern_matching --> e'
------------------------------------------------------- :: match_success
|- match v with pattern_matching --> e'

---------------------------------------------------------------- :: and
|- e1 && e2 --> if e1 then e2 else false

---------------------------------------------------------------- :: or
|- e1 || e2 --> if e1 then true else e2

---------------------------------------------------------------- :: while
|- while e1 do e2 done --> if e1 then (e2; while e1 do e2 done)

{{ com
We specify the evaluation of $[[e1]]$ before $[[e2]]$ in $\ottkw{for}$ loops,
which appears to follow the implementation.
}}

|- e1 -->L e1'
------------------------------------------------------- :: for_ctx1
|- for x = e1 for_dirn e2 do e3 done -->L for x = e1' for_dirn e2 do e3 done

|- e2 -->L e2'
------------------------------------------------------- :: for_ctx2
|- for x = v1 for_dirn e2 do e3 done -->L for x = v1 for_dirn e2' do e3 done

intn1 <= intn2
------------------------------------------------------- :: for_to_do
|- for x = intn1 to intn2 do e done --> (let x = intn1 in e); for x = intn1 .+ 1 to intn2 do e done

intn1 > intn2
------------------------------------------------------- :: for_to_done
|- for x = intn1 to intn2 do e done --> ()

intn2 <= intn1
------------------------------------------------------- :: for_downto_do
|- for x = intn1 downto intn2 do e done --> (let x = intn1 in e); for x = intn1 .- 1 downto intn2 do e done

intn2 > intn1
------------------------------------------------------- :: for_downto_done
|- for x = intn1 downto intn2 do e done --> ()

{{ com
We specify right-to-left evaluation order for tuples, applied variant
constructors, and $\ottkw{::}$.
}}

|- e -->L e'
------------------------------------------------------- :: tuple_ctx
|- e1,..,em,e,v1,..,vn -->L e1,..,em,e',v1,..,vn

|- e -->L e'
------------------------------------------------------- :: constr_ctx
|- constr (e1,..,em,e,v1,..,vn) -->L constr (e1,..,em,e',v1,..,vn)

|- e -->L e'
------------------------------------------------------- :: cons_ctx1
|- e0 :: e -->L e0 :: e'

|- e -->L e'
------------------------------------------------------- :: cons_ctx2
|- e :: v -->L e' :: v

>>
 <<

{{ com
We specify right-to-left evaluation for records.  The bytecode implementation
appears to go right to left after first reordering the record to correspond to
the field ordering in the record type definition.
}}

{{ com
\ottbreakconclusionlinetrue
}}

%%WIDTH: >~a4/landscape/normalsize
|- expr -->L expr'
------------------------------------------------------- :: record_ctx
|- {fn1=e1;...;fnm=em; field_name=expr; fn1'=v1;...;fnn'=vn} -->L {fn1=e1;...;fnm=em; field_name=expr'; fn1'=v1;...;fnn'=vn}

{{ com
\ottbreakconclusionlinefalse
}}

{{ com
\ottbreakconclusionlinetrue
}}

%%WIDTH: >>a4/landscape/normalsize
|- e -->L e'
------------------------------------------------------- :: record_with_ctx1
|- {v with fn1=e1;...;fnm=em; field_name=e; fn1'=v1;...;fnn'=vn} -->L {v with fn1=e1;...;fnm=em; field_name=e'; fn1'=v1;...;fnn'=vn}

{{ com
\ottbreakconclusionlinefalse
}}

|- e -->L e'
------------------------------------------------------- :: record_with_ctx2
|- {e with field_name1=e1;...;field_namen=en} -->L {e' with field_name1=e1;...;field_namen=en}

{{ com
\ottbreakconclusionlinetrue
}}

%%WIDTH: >>a4/landscape/normalsize
length (v1'')...(vl'') >=1
field_name notin fn1...fnm
------------------------------------------------------- :: record_with_many
|- {{fn1=v1;...;fnm=vm;field_name=v;fn1'=v1';...;fnn'=vn'} with field_name=v'; fn1''=v1'';...;fnl''=vl''} --> {{fn1=v1;...;fnm=vm;field_name=v';fn1'=v1';...;fnn'=vn'} with fn1''=v1'';...;fnl''=vl''}


%%WIDTH: >>a4/landscape/normalsize
field_name notin fn1...fnm
------------------------------------------------------- :: record_with_1
|- {{fn1=v1;...;fnm=vm;field_name=v;fn1'=v1';...;fnn'=vn'} with field_name=v'} --> {fn1=v1;...;fnm=vm;field_name=v';fn1'=v1';...;fnn'=vn'}

{{ com
\ottbreakconclusionlinefalse
}}

|- e -->L e'
------------------------------------------------------- :: record_access_ctx
|- e.field_name -->L e'.field_name

field_name notin fn1...fnn
------------------------------------------------------- :: record_access
|- {fn1=v1;...;fnn=vn;field_name=v;fn1'=v1';...;fnm'=vm'}.field_name --> v

 >>

 <<
defns
JRdefn :: J ::=

defn
|- < definitions , program > labelled_arrow < definitions' , program' > :: :: Rdefn :: defn_
{{ com Definition sequence evaluation }}
{{ tex \vdash \langle [[definitions]] , [[program]] \rangle [[labelled_arrow]] \langle [[definitions']] , [[program']] \rangle }} by
{{ com
Reduces a definition one-step.  Type definitions are moved into
the tuple left sequence as encountered to support typing of intermediate
states.
}}

|- e -->L e'
------------------------------------------------------- :: let_ctx
|- <ds_value, let pat = e ;; definitions> -->L <ds_value, let pat = e' ;; definitions>

|- v matches pat gives <<x1<-v1,..,xm<-vm>>
------------------------------------------------------- :: let_match
|- <ds_value, let pat = v ;; definitions> --> <ds_value, <<x1<-remv_tyvar v1,..,xm<-remv_tyvar vm>>definitions>

letrec_bindings = x1 = function pattern_matching1 and ... and xn = function pattern_matchingn
recfun(letrec_bindings, pattern_matching1) gives e1 ... recfun(letrec_bindings, pattern_matchingn) gives en
------------------------------------------------------- :: letrec
|- <ds_value, let rec letrec_bindings ;; definitions> --> <ds_value, <<x1<-remv_tyvar e1,...,xn<-remv_tyvar en>>definitions>

------------------------------------------------------- :: type
|- <ds_value, type_definition ;; definitions> --> <ds_value ;; type_definition, definitions>

 >>
<<

defns
JSlookup :: JS ::=

defn
store ( location ) gives expr :: :: lookup :: stlookup_ {{ com Store lookup }} by

{{ com
Gets the value stored at a given location.
}}

st(l) gives e'
l noteq l'
------------------------------------------------------- :: rec
st,l'|->e(l) gives e'

------------------------------------------------------- :: found
st,l|->e(l) gives e

defns
JRstore :: JR ::=

defn
|- store labelled_arrow store' :: :: store :: store_ {{ com Store transition }} by

{{ com
Coordinates a store with a label.
}}

------------------------------------------------------- :: empty
|- st --> st

st(l) gives v
------------------------------------------------------- :: lookup
|- st -->(!l=v) st

st'(l) unallocated
------------------------------------------------------- :: assign
|- st,l|->expr,st' -->(l:=v) st,l|->remv_tyvar v,st'

st(l) unallocated
------------------------------------------------------- :: alloc
|- st -->(ref v=l) st,l|->remv_tyvar v



>>
 <<
defns
JRtop :: JR ::=

defn
|- < definitions , program , store > --> < definitions' , program' , store' > :: :: top :: top_
 {{ tex \vdash \langle [[definitions]] , [[program]] , [[store]] \rangle [[-->]] \langle [[definitions']] , [[program']] , [[store']] \rangle }}
 {{ com Top-level reduction }} by

{{ com
The semantics of a machine is described as the parallel evolution of a
structure body (the program) and a store. Each program evaluation step
labelled $[[L]]$ must be matched by a store evaluation step with the
same label.
}}

|- store -->L store'
|- <definitions_value, program> -->L <definitions, program'>
-------------------------------------------------------------------------------------- :: defs
|- <definitions_value, program, store> --> <definitions, program', store'>

 >>
<<


defns
Jebehaviour :: JRB_ ::=
defn
|- expr behaves :: :: ebehaviour :: ebehaviour_ {{ com Expression behaviour }} by

{{ com
This relation describes expressions whose behaviour is defined. This includes
values and expressions that reduce. An expression with no
behaviour is said to be stuck.  In this definition of expression behaviour, we
treat any reducing expression as behaving, no matter what (satisfiable)
constraint is imposed on the label.
}}

---------------------------------------------------------------- :: value
|- v behaves

|- e -->L e'
---------------------------------------------------------------- :: reduces
|- e behaves




>>
 <<

defns
Jdbehaviour :: JRB_ ::=

defn
|- < definitions , program , store > behaves :: :: dbehaviour :: behaviour_
{{ tex \vdash \langle [[definitions]] , [[program]] , [[store]] \rangle [[behaves]] }}
{{ com structure body behaviour }} by

{{ com
As for expressions, a definition sequence behaves if it is a value or if it
reduces (under any label).
}}

---------------------------------------------------------------- :: value
|- <definitions_value, , store> behaves

|- <definitions_value, program, store> --> <definitions', program', store'>
---------------------------------------------------------------- :: reduces
|- <definitions_value, program, store> behaves

 >>
<<


{{ com
\section{Statistics}
\label{sec.statistics}

}}


embed
{{ coq
Hint Constructors JM_matchP JM_match
                  Jrecfun Jfunval JRuprim JRbprim
                  JRmatching_step JRmatching_success
                  JR_expr JRdefn
                  JSlookup JRstore
                  JRB_ebehaviour
                  : rules.
}}
 embed
 {{ coq
 Hint Constructors JRdefn JRtop JRB_dbehaviour : rules.
 }}